stfuifinHinHifuuHUfiuu  j 


UC-NRLF 


$B    S5b,T3D 


I 


■ll|iii, 


LIBRARY 

OF  THE 

University  of  California. 


GIF^T  OK 


Class 


MECHANICS-PROBLEMS 


FOR   ENGINEERING   STUDENTS 


BY 

FRANK-  B.  SANBORN, 

Professor  of  Civil  Engineering  in  Tufts  College 


NEW   YORK: 

THE   ENGINEERING   NEWS   PUBLISHING  COMPANY 

1902 


Copyright,  1902, 

BY 

FRANK  BERRY  SANBORN 


PREFACE 


This  book  contains  many  problems  similar  to  those 
that  are  found  in  text-books ;  but  besides  many  have 
been  developed  from  actual  engineering  conditions.  My 
object  has  been  to  correlate  more  closely  these  every- 
day practical  examples  with  the  important  subjects  in 
Mechanics.  As  an  aid  to  this  end,  fifteen  illustrations 
and  forty-seven  line  cuts  have  been  introduced.  Although 
the  book  is  not  intended  to  take  the  place  of  text-books 
or  lecture  notes,  it  is  hoped  that  it  will  in  many  instances 
supersede  the  hektograph  or  cyclostile  sheets  which  many 
instructors  now  find  it  necessary  to  prepare  and  issue. 

From  my  own  experience  as  student,  engineer  in  prac- 
tice, and  college  instructor,  I  am  convinced  that  the  main 
object  of  a  course  in  Mechanics  should  be  to  prepare  the 
students  to  solve  its  problems.  Therefore  I  have  en- 
deavored to  make  these  problems  —  five  hundred  in 
number  —  fulfill  the  requirements  for  thorough  and  inter- 
esting instruction.  They  have  been  arranged  in  the 
following  order :  Work,  Force,  Motion.  This  order  I 
believe  to  be  on  the  whole  the  most  satisfactory;  but 
some  instructors  prefer  to  teach  first  the  subject  of  Force ; 
others  begin  with  Motion.  To  meet  these  varying  de- 
mands and  other  general  uses  an  alphabetical  classifi- 
cation of  problems  has  been  prepared,  which  makes  it 


2          •  PREFA  CE. 

possible  to  find  any  single  problem,  or  a  collection  of 
problems  that  pertain  to  a  given  subject. 

At  the  beginning  of  each  important  section  of  the 
book  one  problem  is  solved  so  as  to  explain  the  method 
of  solving  similar  problems,  and  to  serve  as  a  guide  Tor 
solutions  to  be  put  in  note-books.  I  believe  that  students 
learn  the  subject  best  and  analyze  most  carefully  the 
steps  taken  by  thus  keeping  this  set  of  college  notes 
unabbreviated  and  in  plain,  concise  form.  In  my  own 
classes  I  require  with  considerable  firmness  that  these 
notes  shall  be  well-kept. 

Besides  the  problems  taken  from  engineering  practice 
valuable  ones  have  been  selected  from  the  following 
examination  papers :  Science  and  Art  (England),  Wool- 
wich (England),  Annapolis  (United  States),  West  Point 
(United  States),  and  Harvard  (United  States);  and  some 
taken  in  whole  or  in  part  from  text-books  on  Mechanics 
by  Thornton,  Lodge,  Perry,  Goodman,  Goodeve  (England), 
Wright,  Hoskins,  Johnson,  Bowser  (United  States). 

Photographs  or  electroplates  have  been  furnished  for 
certain  of  the  illustrations  as  follows : 

Page  17  by  Otto  Gas  Engine  Works;  pages  20  and  32,  Pellon 
Water  Wheel  Company;  page  24,  Wellington-Wild  Coal  Company; 
page  25,  Harrisburg  Foundry  and  Machine  Company;  page  29, 
Fall  River  Iron  Works  Company ;  page  35,  Associated  Factory 
Mutual  Fire  Insurance  Companies  ;  page  63,  Maryland  Steel  Com- 
pany ;  page  64,  Bucyrus  Company  ;  page  112,  A.  J.  Lloyd  &  Co. 

FRANK   B.  SANBORN. 
Turrs  College,  Mass., 
September,  1902. 


CONTENTS 

I.    WORK. 

Problems  i  to  172. 
FOOT-POUNDS  PAGE 

Raising  weights,  overcoming  resistances  of  railroad 
trains,  macliine  punch,  construction  of  wells  and 
chimneys,  operation  of  pumping  engines.  Force  and 
distance  or  foot-pounds  required  in  cases  of  pile- 
driver,  horse,  differential  pulley,  tackle,  tram  car       .     .       7 

HORSE-POWER 

Required  by  windmills,  planing  machines,  gas  engine, 
locomotive,  steam  engines  —  simple,  compound,  triple, 
slow  speed,  high  speed  engines.  Horse-power  from 
indicator  cards,  required  by  electric  lamps,  driving 
belts,  steam  crane,  coal  towers,  pumping  engine, 
canals,  streams,  turbines,  water-wheels.  Efficiency, 
force  or  distance  required  in  cases  of  fire  pumps, 
mines,  bicycles,  shafts,  railroad  trains,  air  brakes,  the 
tide,  electric  motors,  freight  cars,  ships 16 

ENERGY 

Foot-pounds,  horse-power,  velocity:  —  Ram,  hoisting- 
engine,  blacksmith,  electric  car,  -bullet,  cannon,  nail, 
pendulum.  Energy  resulting  from  motion  of  fly-wheel 
and  energy  required  by  jack-screw 43 


4  CONTENTS. 

II.    FORCE. 

Problems  172  to  405. 

FORCES  ACTING   AT   A    POINT  page 

Canal  boat  being  towed,  rods,  struts,  beams,  derrick, 
cranes  set  as  in  action ;  balloon  held  by  rope,  ham- 
mock supported  ;  wagon,  trucks,  picture  supported  ; 
forces  in  frames  of  car  dumper,  tripod,  shear  legs, 
dipper  dredge  ;  also  in  triangle,  square,  sailing  vessel, 
rudder,  foot-bridge,  roof- truss 51 

MOMENTS    FOR    PARALLEL   FORCES 

Beam  balanced,  pressure  on  supports,  propelling  force 
,  of  oars,  raising  anchor  force  at  capstan,  bridge  loaded 
pressure  on  abutments,  lifting  one  end  of  shaft,  boat 
hoisted  on  davit,  forces  acting  on  triangle,  square, 
supports  of  loaded  table  and  floor        72 

COUPLES 

Brake  wheel,  forces  acting  on  square 84 

STRESSES 

Beam  leaning  against  wall,  post  in  truss,  rope  pull  on 
chimney,  connecting  rod  of  engines,  trap-door  lield  up 
by  chain 86 

CENTER   OF  GRAVITY 

Rods  with  loads,  metal  square  and  triangle,  circular 
disk  with  circular  hole  punched  out,  box  with  cover 
open,  rectangular  plane  with  weight  on  one  end, 
irregular  shapes,  solid  cylinder  in  hollow  cylinder, 
cone  on  top  of  hemisphere 90 

FRICTION 

Weight  moved  on  level  table,  stone  on  ground, 
block  on  inclined  plane,  gun  dragged  up  hill,  cone 
sliding  on  inclined  plane  ;  friction  of  planing  machine. 


CONTENTS.  5 

PAGE 

locomotives,  trains,  ladder  against  wall,  bolt  thread, 
rope  around  a  post ;  belts,  pulleys  and  water-wheels 
in  action  ;  heat  generated  in  axles  and  bearings.      .    .    96 


III.    MOTION. 

Problems  405  to  500. 

UNIFORM    ACCELERATION 

Railroad  train,  ice  boat,  stone  falling  and  depth  of 
well,  balloon  ascending,  cable  car  running  wild.   .     .     .111 

RELATIVE   VELOCITY 

Aim  in  front  of  deer,  rowing  across  river,  bullet  hit- 
ting balloon  ascending,  rain  on  passenger  train,  wind 
on  steamer,  two  passing  railroad  trains 117 

DISTANCE,  V£L0CITY,  FRICTION,  ANGLE  OF 
INCLINATION 
Train  stopped,  steamer  approaching  dock,  cannon 
recoil,  locomotive  increasing  speed,  body  moved  on 
table,  box-machine,  motion  of  table,  barrel  of  flour  on 
elevator,  man's  weight  on  elevator,  cage  drawn  up 
coal  shaft 119 

PROJECTILES 

Inclination  for  bullet  to  strike  given  point,  motion 
down  plane,  stone  dropped  from  train,  thrown  from 
tower,  projectile  from  hill,  from  bay  over  fortification 
wall 125 

PENDULUMS 

Simple,  conical,  ball  in  passenger  car 129 

IMPACT 

Water  suddenly  shut  off,  cricket  ball  struck,  hammer 
falling  on  pile,  shot  from  gun,  bullet  from  rifle,  freight 
and  passenger  trains  collide 130 


MECHANICS-PROBLEMS 

I.  WORK 

FOOT-POUNDS 

1.  A  train  weighing  lOO  tons  moves  30  miles  an 
hour  along  a  horizontal  road  ;  the  resistances  are  8 
pounds  per  ton.  Find  the  quantity  of  work  expended 
each  hour. 

Work  =  force  x  distance 
Force  =  8  x  100 

=  800  pounds 
Distance  =  30  X  5  280 
=  158  400  feet 
.'.  Work  =  800  pounds  X  158  400  feet 

=  126  720  000  foot-pounds  each  hour. 

2.  Find  the  work  done  by  an  engine  in  drawing  a 
train  one  mile  along  a  level  railway,  when  the  con- 
stant resistances  of  friction,  air,  and  so  on,  are  one 
ton. 

3.  A  hole  is  punched  through  a  plate  of  wrought- 
iron  one-half  inch  in  thickness,  and  the  pressure  ope- 
rating the  punch  is  estimated  at  36  tons.  Assuming 
that  the  resistance  to  the  punch  is  uniform,  find  the 
number  of  foot-pounds  of  work  done. 

4.  Find  what  work  is  being  done  per  minute  — 
that  is,  find  the  activity  or  the  power  of  a  pumping 

7 


8  MECHANICS-PROBLEMS. 

engine  which  is  raising   2  000   gallons  of    water  an 
hour  from  a  mine  300  feet  deep. 

5.  If  a  weight  of  i  130  pounds  be  lifted  up  20 
feet  by  20  men  twice  in  a  minute,  how  much  work 
does  each  man  do  per  hour  ? 

6.  A  number  of  men  can  each  do,  on  the  average, 
495  000  foot-pounds  of  work  per  day  of  8  hours. 
How  many  such  men  are  required  to  work  at  the  rate 
of  10  horse-power — 33000  x  10  foot-pounds  per 
minute } 

7.  It  is  said  that  a  horse  can  do  about  1 3  200  000 
foot-pounds  of  work  in  a  day  of  8  hours,  walking  at 
the  rate  of  2\  miles  per  hour.  What  pull  in  pounds 
could  such  a  horse  exert  continuously  during  the 
working-day } 

8.  The  surface  of  the  water  in  a  well  is  at  a  depth 
of  20  feet  from  the  surface  of  the  ground,  and  when 
500  gallons  have  been  pumped  out  the  surface  is  low- 
ered to  26  feet.  Find  the  number  of  units  of  work 
done  in  the  operation. 

9.  One  of  the  largest  chimneys  in  America  is  that 
of  the  Clark  Thread  Co.  at  Newark,  N.J.  Its  height 
is  335  feet,  interior  diameter  11  feet,  outside  diameter 
at  base  28^^  feet,  at  top  14  feet.  Find  the  work  done 
in  raising  the  material  from  the  ground  to  its  place  in 
the  chimney. 

10.  A  chain  hanging  vertically  520  feet  long,  weigh- 
ing 20  pounds  per  foot,  is  wound  up.  What  work  is 
done .'' 


WORK—  FO  O  T-PO  UNDS.  9 

11.  A  chain  of  weight  300  pounds  and  length  150 
feet,  with  a  weight  of  500  pounds  at  the  end  of  it,  is 
to  be  wound  up  by  a  capstan.  What  work  will  be 
done  .-* 

12.  A  stream  of  water  is  20  feet  wide,  its  average 
depth  is  3  feet,  and  the  average  velocity  in  the  cross- 
section  is  3  miles  per  hour.  If  there  is  an  available 
fall  of  200  feet,  how  much  potential  energy  is  possessed 
by  the  quantity  flowing  each  minute  .?  ( Weight  of 
water  may  be  taken  as  62.5  pounds  per  cubic  foot.) 

13.  A  horse  draws  150  pounds  of  earth  out  of  a 
well,  by  means  of  a  rope  going  over  a  fixed  pulley, 
which  moves  at  the  rate  of  2\  miles  an  hour.  Neg- 
lecting friction,  how  many  units  of  work  does  this 
horse  perform  a  minute  t 

14.  A  cylindrical  shaft  14  feet  in  diameter  must  be 
sunk  to  a  depth  of  10  fathoms  through  chalk,  the 
weight  of  which  is  144  pounds  per  cubic  foot.  Find 
the  work  done. 

15.  A  well  is  to  be  dug  20  feet  deep  and  4  feet  in 
diameter.  Find  the  work  in  raising  the  material,  sup- 
posing that  a  cubic  foot  of  it  weighs  140  pounds. 

16.  A  horse  draws  earth  from  a  trench  by  means 
of  a  rope  going  over  a  pulley.  He  pulls  up,  twice 
every  5  minutes,  a  man  weighing  1 30  pounds,  and  a 
barrowful  of  earth  weighing  260  pounds.  Each  time 
the  horse  goes  forward  40  feet.  Find  the  useful 
work  done  per  hour. 

17.  A  body  weighing  50  pounds  slides  a  distance 


10  MECHANICS-PROBLtlMS. 

of  8  feet  down  a  plane  inclined  20°  to  the  horizon- 
tal, against  a  constant  retarding  force  of  4  pounds. 
Compute  the  total  work  done  upon  the  body  by 
(gravity)  its  weight  and  the  friction. 

18.  What  electrical  current  expressed  in  amperes 
will  be  used  by  a  250-volt  electric  hoist  when  raising 
2  500  pounds  of  coal  per  minute  from  a  ship's  hold 
150  feet  below  dump  cars  on  trestle  work,  the  effi- 
ciency of  the  whole  arrangement  being  50  per  cent } 

(  I  horse-power  =  746  watts    |^ 
(  Watts      =  volts  X  amperes  S 

19.  The  hammer  of  a  pile-driver,  weighing  500 
pounds,  is  raised  to  a  height  of  20  feet  and  then 
allowed  to  fall  upon  the  head  of  a  pile,  which  is  driven 
into  the  ground  1  inch  by  the  blow.  Find  the  aver- 
age force  which  the  hammer  exerts  upon  the  head  of 

the  pile. 

Work  =  force  X  distance 
=  500  X  20 
=  10  000  foot-pounds 
Distance  =  j^  ^^^^ 
.'.  10  000  foot-pounds  =  force  X  yV  ^^^^ 
.'.  force  =  10  000  X  12 

=  120  000  pounds 

20.  A  hammer  weighing  i  ton  falls  from  a  height 
of  24  feet  on  the  end  of  a  vertical  pile,  and  drives  it 
half  an  inch  deeper  into  the  ground.  Assume  the 
driving  force  of  the  hammer  on  the  pile  to  be  con- 
stant while  it  lasts,  and  find  its  amount  expressed  in 
tons  weight. 


V/OKK  —  FO  O  r-PO  UNDS.  I  I 

21.  What  energy  is  stored  in  a  cross-bow  whose 
cord  has  been  pulled  15  inches  with  a  maximum 
force  of  224  pounds  ? 

22.  A  train  of  150  tons  is  running  at  60  miles  an 
hour.  What  force  is  required  to  stop  it  in  a  quarter 
of  a  mile  } 

23.  A  railway  car  of  4  tons,  moving  at  the  rate  of 
5  miles  an  hour,  strikes  a  pair  of  buffers  which  yield 
to  the  extent  of  6  inches.  Find  the  average  force 
exerted  upon  them. 

24.  If  25  cubic  feet  of  water  are  pumped  every  5 
minutes  from  a  mine  140  fathoms  deep,  what  amount 
of  work  is  expended  per  minute  t 

25.  In  pumping  i  000  gallons  from  a  water-cistern 
with  vertical  sides  the  surface  of  the  water  is  lowered 
5  feet.  Find  the  work  done,  the  discharge  being  10 
feet  above  the  original  surface. 

26.  A  pumping-engine  is  partly  worked  by  a  weight 
of  2  tons,  which  at  each  stroke  of  the  pump  falls 
through  4  feet  ;  the  pump  makes  10  strokes  a  min- 
ute. How  many  gallons  of  water  (one  gallon  weigh- 
ing 8^  pounds)  are  lifted  per  minute  by  the  weight 
from  a  depth  of  200  feet .? 

Work  =  force  X  distance 
Force  =  2X2  000  X  10 

=  40  000  pounds 
Distance  =  4  feet 

Work  =  40  000  pounds  x  4  feet 

=  160  000  foot-pounds 


1 2  MECHANICS-rKOBLEMS. 

To  find  the  number  of  gallons  of  water  that  can 
be  lifted  by  this  amount  of  work : 

Work  =  force  X  distance 

1 60  000  foot-pounds  =  force  x  200  feet 

.               160  000 
force  =  

200 

=  800  pounds 
=  96  gallons 

27.  A  uniform  beam  weighs  i  000  pounds,  and  is 
20  feet  long;  it  hangs  by  one  end,  round  which  it  can 
turn  freely.  How  many  foot-pounds  of  work  must 
be  done  to  raise  it  from  its  lowest  to  its  highest 
position  1 

28.  A  weight  of  200  pounds  is  to  be  raised  to  a 
height  of  40  feet  by  a  cord  passing  over  a  fixed  smooth 
pulley ;  it  is  found  that  a  constant  force  P,  pulling 
the  cord  at  its  other  end  for  three-fourths  of  the  as- 
cent, communicates  sufficient  velocity  to  the  weight 
to  enable  it  to  reach  the  required  height.      Find  P. 

29.  A  horse  drawing  a  cart  along  a  level  road  at 
the  rate  of  2  miles  per  hour  performs  29  216  foot- 
pounds of  work  in  3  minutes.  What  pull  in  pounds 
does  the  horse  exert  in  drawing  the  cart  } 

30.  A  body  weighing  10  pounds  slides  down  an 
inclined  plane  whose  height  is  25  feet  ;  it  reaches 
the  foot  of  the  plane  with  a  velocity  of  30  feet  per 
second.  During  the  motion  how  many  foot-pounds 
of  energy  have  been  expended  on  friction  and  other 
resistances  1 

(The  velocity  of  a  body  falling  unimpeded  is  z'=  y/2gh.) 


WORK—  FOO  T-PO  UNDS.  1 3 

31.  If  a  horse  walking  once  round  a  circle  10  yards 
across  raises  a  ton  weight  18  inches,  what  force  does 
he  exert  over  and  above  that  necessary  to  overcome 
friction  ? 

32.  If,  neglecting  frictions,  a  power  of  10  pounds, 
acting  on  an  arm  2  feet  long,  produces  in  a  screw- 
press  a  pressure  of  half  a  ton,  what  would  be  the 
pitch  of  the  screw  } 

33.  What  is  the  ratio  of  the  weight  to  the  power, 
in  a  screw-press  working  without  friction,  when  the 
screw  makes  4  turns  in  the  inch,  and  the  arm  to 
which  the  power  is  applied  is  2  feet  long  ? 

34.  What  force  applied  at  the  end  of  an  arm 
18  inches  long  will  produce  a  pressure  of  i  000 
pounds  upon  the  head  of  a  smooth  screw  when  1 1 
turns  cause  the  head  to  advance  two-thirds  of  an 
inch  } 

35.  Find  the  mechanical  advantage  in  a  differential 
screw,  if  the  length  of  the  power  arm  is  2  feet,  and 
there  are  4  threads  to  the  inch  in  the  large  screw, 
and  5  threads  to  the  inch  in  the  small  screw. 

36.  In  a  differential  pulley,  if  the  radii  of  the 
pulleys  in  the  fixed  block  are  as  3  to  2  ;  and  if  the 
weight  of  the  lower  block  is  ij  pounds,  what  weight 
can  be  raised  by  a  force  of  5  pounds } 

37.  In  a  wheel  and  axle  the  diameter  of  the  wheel 
is  7  feet,  of  the  axle  7  inches.     What  weight  can  be 


14 


MECHA  NICS-PROBLEMS. 


raised  by  a  force  of   lo  pounds  acting  at  the  circum- 
ference of  the  wheel  ? 

38.  A  weight  of  448  pounds  is  raised  by  a  cord 
which  passes  round  a  drum  3  feet  in  diameter,  having 
on  its  shaft  a  toothed  wheel  also  3  feet  in  diameter  ; 
a  pinion  8  inches  in  diameter,  and  driven  by  a  winch 
handle  16  inches  long,  gears  with  the 
wheel.  Find  the  power  to  be  applied  to 
the  winch  handle  in  order  to  raise  the 
weight. 

39.  A  tackle  is  formed  of  two  blocks, 
each  weighing  1 5  pounds,  the  lower  one 
being  a  single  movable  pulley,  and  the 
upper  or  fixed  block  having  two  sheaves ; 
the  parts  of  the  cord  are  vertical,  and 
the  standing  end  is  fixed  to  the-  movable 
block.  What  pull  on  the  cord  will  sup- 
port 200  pounds  hung  from  the  movable 

block.?  and  what  will  then  be  the  pull  on  the  staple 

at  the  upper  block  '^. 

40.  A  weight  of  400  pounds  is  being  raised  by  a 
pair  of  pulley  blocks,  each  having  two  sheaves  ;  the 
standing  part  of  the  rope  is  fixed  to  the  upper  block, 
and  the  parts  of  the  rope,  whose  weight  may  be  dis- 
regarded, are  considered  to  be  vertical  ;  each  block 
weighs  10  pounds.  What  is  the  pressure  on  the 
point  from  which  the  upper  block  hangs  .? 

41.  Two  equal  weights,  each  1 1 2  pounds,  are  joined 
by  a  rope  which  runs  over  two  pulleys  A  and  B  12 


Fig.  I. 


WORK—  FO  O  T-PO  UNDS.  I  5 

feet  apart  and  in  the  same  horizontal  hne.  If  a 
weight  of  lo  pounds  is  attached  to  the  rope  half-way 
between  A  and  B,  find  the  distance  in  inches  to  which 
the  rope  is  deflected  below  the  level  of  A  B. 

42;  A  Aveight  of  500  pounds,  by  falling  through 
36  feet,  lifts,  by  means  of  a  machine,  a  weight  of  60 
pounds  to  a  height  of  200  feet.  How  many  units  of 
work  have  been  expended  on  friction,  and  what  pro- 
portion does  the  expenditure  bear  to  the  whole 
amount  of  work  done  ? 

43.  The  pull  on  a  tram-car  was  registered  when  the 
car  was  at  the  following  distances  along  the  track  : 
o,  200  pounds;  10  feet,  150  pounds;  25  feet,  160 
pounds;  32  feet,  156  pounds;  41  feet,  163  pounds; 
56  feet,  170  pounds;  60  feet,  165  pounds;  73  feet, 
160  pounds.  What  is  the  average  (space)  pull  on  the 
car,  and  what  is  the  effective  work  done  in  pulling 
the  car  through  the  distance  of  73  feet } 

44.  In  lifting  an  anchor  of  i|  tons  from  a  depth 
of  1 5  fathoms  in  6  minutes,  what  is  the  useful  man- 
power, if  a  man-power  is  defined  as  3  500  foot-pounds 
per  minute } 

45.  Four  hundred  weight  of  material  are  drawn 
from  a  depth  of  80  fathoms  by  a  rope  weighing  1.15 
pounds  per  linear  foot.  How  much  work  is  done 
altogether,  and  how  much  per  cent  is  done  in  lifting 
the  rope.''  How  many  units  of  33000  foot-pounds 
per  minute  would  be  required  to  raise  the  material  in 
4|  minutes } 


1 6  MECHANICS-PROBLEMS 


HORSE-POWER 


46.  A  weight  of  3  tons  is  raised  through  50  feet 
in  a  quarter  of  a  minute.  What  horse-power  must 
be  used  ? 

Work  done  fn  \  minute  =3X2  000  x  50  foot-pounds. 

Work  done  in  i  minute  =4  X  3  X  2  000  x  50  foot-pounds. 

Now,  I  horse-power        =  -XiZ  000  foot-pounds  per  minute. 

...  4X3X2  000  xso 

.-.    required  horse-power  = — 

33000 

=  36^ 

47.  A  man  weighing  1 5  5  pounds  carries  a  weight 
of  65  pounds  to  a  vertical  height  of  20  feet.  How 
many  foot-pounds  of  work  has  he  done }  If  he  make 
20  such  journeys  in  an  hour,  at  what  rate  in  horse- 
power does  he  work } 

48.  A  windmill  raises  by  means  of  a  pump  22  tons 
of  water  per  hour  to  a  height  of  60  feet.  Supposing 
it  to  work  uniformly,  calculate  its  horse-power. 

49.  The  travel  of  the  table  of  a  planing-machine 
which  cuts  both  ways  is  9  feet.  If  the  resistance 
while  cutting  be  taken  at  400  pounds,  and  the 
number  of  revolutions  or  double  strokes  per  hour  be 
80,  find  the  horse-power  absorbed  in  cutting. 

50.  A  forge  hammer  weighing  300  pounds  makes 
100  lifts  a  minute  ;  the  perpendicular  height  of  each 
lift  is  2  feet.  What  is  the  horse-power  of  the  engine 
that  operates  20  such  hammers  t 


WORK—  HORSE-PO  WER. 


17 


51.  What  would  be  the  indicated  horse-power  of  a 
gas  engine  which  has  a  piston  1 2  inches  in  diameter 
and  a  crank  8  inches  long  ?     The  engine  working  at 


1 8  MECHANICS-PROBLEMS. 

150  revolutions  a  minute,  there  being  an  explosion 
every  2  revolutions  and  the  mean  effective  pressure 
in  the  cylinder  during  a  cycle  being  62  pounds  per 
square  inch. 

52.  How  many  horse-power  would  it  take  to  raise 
3  hundred  weight  of  coal  a  minute  from  a  pit  whose 
depth  is  660  feet  t 

53.  Find  the  horse-power  of  an  engine  which  is  to 
raise  30  cubic  feet  of  water  per  minute  from  a  depth 
of  440  feet. 

54.  Find  the  horse-power  required  to  draw  a  train 
of  100  tons,  at  the  rate  of  30  miles  an  hour,  along  a 
level  railroad,  the  resistance  from  friction  being  16 
pounds  per  ton. 

55.  Each  of  the  two  cylinders  in  a  locomotive 
engine  is  16  inches  in  diameter  and  the  length  of 
crank  is  i  foot.  If  the  driving-wheels  make  105 
revolutions  per  minute,  and  the  mean  effective  steam- 
pressure  is  85  pounds  per  square  inch,  what  is  the 
horse-power  t 

56.  The  weight  of  a  train  is  95.5  tons,  and  the 
drawbar  pull  is  6  pounds  per  ton.  Find  the  horse- 
power required  to  keep  the  train  running  at  25  miles 
per  hour. 

57.  A  train,  whose  weight  including  the  engine  is 
100  tons,  is  drawn  by  an  engine  of  150  horse-power ; 
friction  is  14  pounds  per  ton  —  all  other  resistances 
neglected.  Find  the  maximum  speed  which  the 
engine  is  capable  of  maintaining  on  a  level  track. 


WORK—HOKSE-POWER.  19 

58.  A  dynamo  is  driven  by  an  engine  that  develops 
230  horse-power.  If  the  efficiency  of  dynamo  is  0.81 
what  ''  activity "  in  kilowatts  is  represented  by  the 
current  generated } 

(i  kilowatt  =  1.340  horse-power.) 

59.  Electric  lamps  giving  i  candle-power  for  4 
watts  ia)  how  many  10-  and  {b)  how  many  i6-candle 
lamps  may  be  worked  per  electric  horse-power  t  The 
combined  efficiency  of  engine,  dynamo,  and  gearing 
being  70  per  cent,  what  is  the  candle-power  avail- 
able for  every  indicated  horse-power  .? 

60.  The  section  of  a  stream  is  1 2  square  feet,  the 
average  velocity  of  the  water  is  2  feet  per  second  ; 
there  is  an  available  fall  of  25  feet  ;  what  is  the  horse- 
power available  t  A  turbine  here  drives  a  dynamo  ma- 
chine which  sends  electric  power  to  a  motor  at  a  dis- 
tance. The  efficiency  of  the  turbine  is  70  per  cent ; 
of  the  dynamo,  87  per  cent ;  10  per  cent  of  the  en- 
ergy from  the  dynamo  is  wasted  in  transmission  and 
the  efficiency  of  the  motor  is  72  per  cent.  How 
much  power  is  given  out  by  the  motor }  The  volt- 
age of  the  dynamo  is  102.  What  is  the  current  in 
amperes  t 

61.  A  500-volt  electric  motor  imparts    velocity  to 
an   8-ton   car  so  that  at  the  end  of  20  seconds  it  is^ 
moving,  on  a  level  track  at  the  rate  of   10  miles  an 
hour  ;  the  total  efficiency  of  the  motor  and  car  is  60 
percent.     What  amperes  are  necessary  .? 


20 


MECHANICS-PROBL  EMS. 


62.  A  water-motor  is  driven  by  two  jets  i  inch  in 
diameter,  flowing  with  velocity  of  80  feet  per  second. 
Theoretic  horse-power  would  be  9.9  ;  and  if  efficiency 
of  wheel  is  85  per  cent,  and  the  generator  which 
the  wheel  drives  also  85  per  cent,  what  power  in 
kilowatts  is  represented  by  the  current  that  is 
produced  ? 

63.  What  is  the  necessary  difference  of  tensions  in 
a  driving-belt  30  inches  wide,  which  is  running  4  200 
feet  a  minute  and  transmitting  300  horse-power  1 

64.  Find  the  speed  of  a  driving-pulley  3.5-feet  in 
diameter  to  transmit  6  horse-power,  the  driving-force 
of  the  belt  being  150  pounds. 


WORK—  HORSE-PO  WER.  2 1 

65.  A  belt  can  stand  a  pull  of  lOO  pounds  only. 
Find  the  least  speed  at  which  it  can  be  driven  to 
transmit  20  horse-power. 

66.  A  pulley  3  feet  6  inches  in  diameter,  and  mak- 
ing 150  revolutions  a  minute,  drives  by  means  of  a 
belt,  a  machine  which  absorbs  7  horse-power.  What 
must  be  the  width  of  the  belt  so  that  its  greatest  ten- 
sion may  be  70  pounds  per  inch  of  width,  it  being 
assumed  that  the  tension  in  the  driving-side  is  twice 
that  on  the  slack  side  } 

67.  An  endless  cord  stretched  and  running  over 
grooved  pulleys  with  a  linear  velocity  of  3  000  feet 
per  minute,  transmits  five  horse-power.  Find  the 
tension  of  the  cord  in  pounds. 

68.  In  the  transmission  of  power  by  a  rope  the 
wheel  carrying  the  rope  is  14  feet  in  diameter,  and 
makes  30  revolutions  per  minute,  the  tension  of  the 
rope  being  100  pounds.  Find  the  amount  of  power 
transmitted  as  estimated  in  horse-power. 

69.  A  locomotive  engine,  which  can  work  up  to 
100  horse-power,  is  attached  to  a  train,  whose  mass 
(including  the  locomotive  itself)  is  100  tons.  Assum- 
ing the  total  resistance  to  be  constant  and  equivalent 
to  10  pounds  weight  per  ton,  find  the  greatest  speed 
of  the  train  in  miles  per  hour. 

When  traveling  at  this  speed  the  steam  is  shut 
off.  Find  the  distance  and  the  time  in  which  the 
train  would  be  reduced  to  rest  by  the  resistance 
alone. 


2  2  mp:  chanics-pr  oblems. 

70.  A  train  weighing  lOO  tons  runs  at  42  miles  an 
hour  on  a  level  track,  the  resistance  being  8  pounds 
per  ton.  Find  its  speed  up  a  i  per  cent  grade  (i 
foot  rise  in  100  feet  horizontal)  if  the  engine-power 
is  kept  constant. 

71.  In  18,95  a  passenger  engine  on  the  Lake  Shore 
Railroad  made  a  run  of  ^6  miles  at  the  rate  of  73 
miles  an  hour.  Weight  of  train,  250  tons  ;  resistance 
on  level  track,  i  5  pounds  per  ton.  The  engine  was  a 
10- wheeler,  having  drivers  5  feet  8  inches  in  diameter 
and  cylinders  17  x  24  inches.  Show  that  to  develop 
730  horse-power  the  average  effective  cylinder-pres- 
sure must  have  been  about  37  pounds  per  square  inch. 

72.  What  must  be  the  effective  horse-power  of  a 
locomotive  which  moves  at  the  steady  speed  of  35 
miles  an  hour  on  level  rails,  the  weight  of  engine  and 
train  being  120  tons,  and  the  resistance  16  pounds 
per  ton  .-^  What  additional  horse-power  would  be 
necessary  if  the  rails  were  laid  along  a  gradient  of  i 
in  142  ? 

73.  In  example  72  find  in  each  case  how  far  the 
train  would  move  after  steam  was  shut  off,  assuming 
the  above  constant  resistance  and  neglecting  rotary 
motions.  Find  also  the  speed  of  the  train  after  the 
latter  had  moved  over  a  distance  of  i  000  feet  from 
the  point  where  steam  was  shut  off. 

74.  Find  the  total  horse-power  of  two  engines 
which  are  taking  a  train 'of  250  tons  down  a  grade  of 


WORK  —  HORSE-PO  WER.  2  3 

I  in  200  at  60  miles  an  hour,  supposing  the  resistance 
on  the  level  at  this  speed  to  be  35  pounds  a  ton. 

75.  A  train  of  50  tons  moves  up  a  rough  incline  of 
I  in  10,  the  resistance  caused  by  friction  being  16 
pounds  per  ton.  What  horse-power  must  the  engine 
exert  in  order  to  maintain  a  uniform  speed  of  3  miles 
an  hour } 

76.  Find  the  horse-power  of  a  locomotive  which  is 
to  move  at  the  rate  of  20  miles  an  hour  up  an  incline 
which  rises  i  foot  in  100,  the  weight  of  the  locomo- 
tive and  load  being  60  tons,  and  the  resistance  from 
friction  12  pounds  per  ton. 

77.  A  steam-crane,  working  at  3  horse-power,  is 
able  to  raise  a  weight  of  10  tons  to  a  height  of  50  feet 
in  20  minutes.  What  part  of  the  work  is  done  against 
friction }  If  the  crane  is  kept  at  similar  work  for  8 
hours,  how  many  foot-pounds  of  work  are  wasted  on 
friction } 

78.  The  six-master  shown  on  the  next  page  carries 
5  500  tons  of  coal.  It  is  unloaded  by  small  engines 
which  take  up  i  ton  at  each  hoist  ;  average  lift  from 
hold  of  ship  to  top  of  chutes  which  lead  to  cars, 
35  feet;  weight  of  bucket,  i  ton;  2  trips  are  made 
per  minute,  and  25  per  cent  of  power  of  engine  is 
lost  in  friction  and  transmission.  Find  the  horse- 
power of  each  engine  required  when  two  towers  are 
working. 


WORK—  HORSE-POWER. 


25 


The  illustration  of  six-master  on  opposite  page  accompanies 
Problem  78. 

79.  An  average  size  coal  barge  will  carry  i  600 
tons.  ■  If  it  is  unloaded  by  two  simple  direct  engines, 
the  coal  being  hoisted  65  feet  to  an  elevated  hopper 
on  the  wharf,  weight  of  bucket  i  ton,  and  carrying  i 
ton  of  coal,  what  horse-power  of  engines  would  be  re- 
quired to  unload  the  i  600  tons  in  20  hours? 

80.  The  coal  from  the  hopper  is  run  into  a  car 
which  carries  2  tons,  and  goes  down  a  grade  25  feet 
long  in  25  seconds;  it  strikes  a  cross-bar,  or  "stop- 
per," which,  acting  through  a  distance  of  30  feet, 
brings  the  car  to  rest.  What  is  the  average  force  that 
acts } 


81.    The  engine  shown  in  Fig.  4  has  steam  cylin- 
der 1 5  inches  in  diameter ;  length  of  stroke,  1 5  inches ; 


.26 


ME  CHA  NICS-PR  OBLEMS. 


revolutions  per  minute,  275  ;  mean  effective  pressure, 
;j6  pounds  per  square  inch.     Find  the  horse-power. 

82.  The  indicator  cards  illustrated  herewith  were 
taken  from  an  engine  of  the  type  shown  in  problem 
•81,  diameter  of  steam  cylinder  being  14  inches, 
-length  of  stroke  12  inches,  revolutions  per  minute 
.300,  Scale  on  cut  the  mean  ordinates,  which  were 
produced  by  indicator  springs  of  stiffness  40  pounds 
to  an  inch,  and  compute  the  indicated  horse-power  of 
the  engine. 


Fig.   5.    Full  Load  Indication. 

83.  The  indicator  cards  shown  below  were  taken 
from  one  of  the  triple-expansion  pum ping-engines  at 
the  East  Boston  Station  of  the  Metropolitan  Sewerage. 
The  cards  were  from  two  ends  of  a  high-pressure 
cylinder.  Refer  to  the  cards  and  compute  the  indi- 
cated horse-power.  (A  twenty-four  hours'  duty  trial 
of  this  pumping-engine  was  made  January  17-18, 
1901,  by  engineering  students  of  Tufts  College.) 


IVORK  —  HORSE-PO  WER. 


27 


Fig.  6.  Headend.  Card  shown,  one-half  size;  area  of  original,  4.69  square 
inches;  stiffness  of  spring,  50  pounds  per  square  iSich;  length  of  stroke, 
30  inches;  revolutions  per  minute,  84. 


Fig,  7.  Crankend.  Card  shown,  one-half  size;  area  of  original,  4.62  square 
inches;  stiffness  of  spring,  50  pounds  per  square  inch;  length  of  stroke, 
30  inches;  revolutions  per  minute,  84. 

81  The  average  breadth  of  an  indicator  diagram 
for  one  end  of  a  piston  is  1.58  inches,  and  for  the 
other  end  it  is  1.42  inches,  and  i  inch  represents  32 
pounds  per  square  inch.  Piston,  12  inches  diameter  ; 
crank,  i  foot  long;  revokitions  per  minute,  no. 
What  is  the  indicated  horse-power } 

85.  The  cyHnder  of  a  steam-engine  has  an  internal 
diameter  of  3  feet ;  length  of  stroke,  6  feet ;  and  it 
makes  10  strokes  per  minute.  Under  what  effective 
pressure  per  square  inch  would  it  have  to  work  in 
order  that  the  piston  may  develop  125   horse-power? 


28  MECHANICS-PROBLEMS. 

The  illustration  of  triple-expansion  engines  on  opposite  page 
accompanies  Problem  86. 

86.  Four  pairs  of  triple-expansion  steam-engines 
are  used  to  drive  the  cotton  machinery  of  the  largest 
Fall  River  corporation.  One  of  these  engines  shown 
in  illustration  has  cylinders  26^  inches  diameter,  i6\, 
and  54.  The  steam  pressures  are  :  In  main  pipe,  i  50 
pounds  per  square  inch  ;  in  receiver  between  high  and 
intermediate  cylinders,  40  pounds  ;  in  receiver  between 
intermediate  and  low,  5  pounds.  Vacuum  is  27 
inches.  The  mean  effective  pressures  in  the  cylin- 
ders are  respectively  54  pounds  per  square  inch,  23|- 
and  1 2\.  Length  of  stroke  is  5  feet ;  piston  speed, 
660  feet  per  minute.     Calculate  the  horse-power. 

87.  An  engine  is  required  to  drive  an  overhead 
traveling  crane  for  lifting  a  load  of  30  tons  at  4  feet 
per  minute.  The  power  is  transmitted  by  means  of 
2^-inch  shafting,  making  160  revolutions  per  minute. 
The  length  of  the  shafting  is  250  feet ;  the  power  is 
transmitted  from  the  shaft  through  two  pairs  of  bevel 
gears  (efficiency  90*%  each,  including  bearings),  and 
one  worm  and  wheel  (efficiency  85%,  including  bear- 
ings). Taking  the  mechanical  efficiency  of  the  steam- 
engine  at  80%,  calculate  the  required  horse-power  of 
the  engine. 

88.  An  engine  working  at  50  horse-power  is 
driven  by  steam  at  75  pounds  pressure  acting  on  pis- 
tons in  two  cylinders.  If  the  area  of  each  piston  is  72 
square  inches,  and  the  length  of  stroke  2  feet,  how 
many  revolutions  does  the  fly-wheel  make  per  minute  ? 


30  MECHANICS-PROBLEMS.  x^ 

89.  The  steam-engine  in  use  at  the  Worsted 
Weaving  Mill  of  the  Pacific  Mills  at  Lawrence, 
Mass.,  is  a  Corliss  type  cross-compound  with  steam 
cylinders  19  and  36  inches  diameter;  stroke,  42 
inches;  revolutions,  100  per  minute;  mean  effective 
pressures,  60  pounds  and  1 3  pounds.  Find  how  many 
looms  weaving  worsted  dress-goods  said  engine  will 
drive,  each  loom  requiring  \  horse-power. 

90.  A  ship  laden  with  coal  must  be  unloaded  at 
the  rate  of  22  tons  of  coal  in  10  minutes.  If  the 
height  of  lift  is  150  feet,  what  horse-power  of  engines 
will  be  required  } 

91.  The  fuel  used  in  running  a  steam-engine  is 
coal  of  such  composition  that  the  combustion  of  i 
pound  produces  heat  sufficient  to  raise  the  tempera- 
ture of  1 2  000  pounds  of  water  1  °  Fahr.  It  is  found 
that  3 1  pounds  of  fuel  are  consumed  per  horse-power 
per  hour.  What  is  the  efficiency  of  the  entire  appa- 
ratus } 

92.  A  steam-engine  uses  coal  of  such  composition 
that  the  combustion  of  i  pound  generates  10  000 
British  thermal  units.  If  40  pounds  of  coal  are  used 
per  hour,  and  if  the  efficiency  is  0.08,  what  horse- 
power is  realized  } 

93.  The  cylinder  of  a  Corliss-type  steam-engine  is 
30  inches  in  diameter,  stroke  48  inches,  and  it  makes 
85  revolutions  per  minute.  The  steam  pressure  be- 
ing 90  pounds  per  square  inch,  what  is  the  horse- 
power of  the  engine } 


WORK—  HORSE-POWEK.  31 

94.  The  piston  of  a  steam-engine  is  15  inches  in 
diameter  ;  its  stroke  is  2|  feet,  and  it  makes  20  revo- 
lutions per  minute  ;  the  mean  pressure  of  the  steam 
on  it  is  15  pounds  per  square  inch.  How  many  foot- 
pounds of  work  are  done  by  the  steam  per  minute, 
and  what  is  the  horse-power  of  the  engine  t 

95.  An  engine  has  a  6-foot  stroke,  the  shaft  makes 
30  revokitions  per  minute,  the  av^erage  steam  pres- 
sure is  25  pounds  per  square  inch.  Required  the 
horse-power  when  the  area  of  the  piston  is  i  800 
square  inches,  the  modukis  of  the  engine  being  i|. 

96.  The  diameter  of  a  st^am-engine  cylinder  is  9 
inches  ;  the  length  of  crank,,  9  inches  ;  the  number  of 
revolutions  per  minute,  iio;  the  mean  effective  pres- 
sure of  the  steam  35  pounds  per  square  inch.  Find 
the  indicated  horse-power. 

97.  Find  the  horse -power  of  an  engine  which  is 
drawing  120  tons  up  an  incline  of  i  in  300  at  30 
miles  an  hour  against  wind  and  frictional  resistances 
of  20  pounds  a  ton. 

98.  The  area  of  a  cross-section  of  the  Charles  River 
at  Riverside,  Massachusetts,  is  408  square  feet.  The 
velocity  of  current  as  found  by  rod  floats  and  current 
meter,  April  17  and  22,  1902,  was  1.12  feet  per  sec- 
ond. What  would  be  the  theoretic  horse-power  of 
this  quantity  of  water  at  the  Waltham  dam,  which 
gives  a  fall  of  12.58  feet  ? 


32 


ME  CHA  NICS-PK  OBL  EMS. 


The  illustration  on  opposite  page  of  canal  and  mills  at  Manches- 
ter, N.H.  accompanies  Problem  103. 

99.  Find  the 
useful  horse-power 
of  a  water-wheel, 
supposing  the 
stream  to  be  100 
feet  wide  and  5  feet 
deep,  and  to  flow 
with  a  velocity  of 
\  foot  per  second  ; 
the  height  of  the 
fall  is  24  feet,  and 
the  efficiency  of 
the  wheel  70  per 
cent. 

100.    A    small 
Fig.  8. —water-Power.  stream    has    mean 

velocity  of  35  feet  per  minute,  fall  of  13  feet  and  a 
mean  section  of  5  feet  by  2.  On  this  stream  is 
erected  a  water-wheel  whose  modulus  is  0.65.  Find 
the  horse-power  of  the  wheel. 

101.  Given  the  stream  in  example  100,  what  must 
be  the  height  of  the  fall  to  grind  10  bushels  of 
corn  per  hour,  if  the  modulus  of  the  wheel  is  0.4 .? 

102.  How  many  cubic  feet  of  water  must  be  de- 
scending the  fall  per  minute  in  example  100,  in  order 
that  the  wheel  may  grind  at  the  rate  of  28  bushels 
per  hour .'' 


34  ■    :    MECHANICS-PROBLEMS. 

103.  The  illustration  on  the  preceding  page  shows 
the  canal  at  Manchester,  N.H.,  as  it  passes  the  mills 
of  the  Amoskeag  Manufacturing  Company,  and  the 
Manchester  Mills  Company.  Width  is  5  i  feet,  depth 
of  water  8.9  feet,  velocity  of  flow  1.13  feet  per  second. 
What  quantity  of  water  is  flowing }  The  height  of 
fall  for  the  turbines  being  27.3  feet,  what  is  the 
theoretic  horse-power } 

104.  The  mean  section  of  the  Merrimac  Canal  just 
before  it  enters  the  mills  of  the  Merrimac  Manufac- 
turing Company  at  Lowell,  Mass.,  is  48.2  feet  by  10.6 
feet;  mean  velocity  on  Nov.  23,  1901,  was  2.37  feet 
per  second  ;  the  water-wheels  had  a  net  fall  of  ^35-67 
feet,  and  gave  an  efficiency  of  about  J  J  per  cent.  Find 
the  number  of  broad  looms  weaving  cotton  sheetings 
that  may  be  driven  2|  looms  requiring  one  horse- 
power. 

105.  The  estimated  discharge  of  the  nine  turbines 
at  Niagara  Falls  in  1898  was  430  cubic  feet  per  sec- 
ond for  each  turbine.  The  average  pressure  head  on 
the  wheels  was  that  due  to  .a  fall  of  about  136  feet. 
Cbrri^ffetlfSSctual  horse-power'avkilable  -fr^m  ail  tur- 
bines, allowing  an  efficiency  of  82  percejit.      >  ;„ 

106.  The  average  flow  over  Niagara  Falls  is  270  oco 
cubic  feet  per  second.  The  height  of  fall  is  161  feet. 
In  round  numbers  what  horse-powder  is  developed } 

107.  Calculate  the  horse-power  that  can  be  obtained 
for  one  minute  from  an  accumulator  which  makes 


WORK  —  HORSE-PO  WER.  3  5 

one  stroke  in  a  minute  and  has  a  ram  of  20  inches 
diameter,  23  feet  stroke,  loaded  to  a  pressure  of  750 
pounds  per  square  inch. 


Fig.  9.    An  Underwriter  Fire-Pump  with  Standard  Fittings. 

108.  A  fire-pump  for  protection  of  a  50  000-spindle 
cotton-mill  will  deliver  i  000  gallons  of  water  per 
minute  at  100  pounds  pressure.      Large  boiler  capa- 


36  MECHANICS-PROBLEMS. 

city  is  required  for  such  a  fire-pump  and  for  the 
above  size  150  horse-pov\er  would  be  used.  What 
portion  of  this  boiler  capacity  would  be  required  in 
actual  work  of  delivering  water  ? 

I  000  gallons  =  8  355  pounds  per  minute 

100  pounds  per  square  inch  pressure  =  230.4  feet 

elevation 
Work  =  8  355  X  230.4 

=  I  924992  foot-pounds  per  minute 
=  58.3  horse-power 

Portion  of  boiler  used  =  ^-^ 
150 
=  .39  (About  one-third.) 

109.  An  Underwriter  fire-pump  to  protect  a  me- 
dium-sized factory  will  deliver  four  streams  of  water 
through  I  |-inch  smooth  nozzles  with  pressure  at  base 
of  play  pipes  of  50  pounds  per  square  inch.  This 
would  correspond  to  a  discharge  of  i  060  gallons  per 
minute.  Loss  of  pressure  through  nozzle  can  be  neg- 
lected ;  and  loss  in  quantity  of  discharge  by  slippage, 
short  strokage,  and  so  on  will  be  about  10  per  cent. 
Find  the  work  done  by  the  pump. 

110.  A  pump  of  medium  size  used  for  fire  pro- 
tection of  a  factory  will  deliver  three  i  |-inch  fire 
streams  at  80  pounds  pressure.  To  give  ample 
boiler  capacity  70  per  cent  should  be  provided  as 
surplus  capacity.  What  should  be  the  total  boiler 
capacity  } 


WORK  —  HORSE-PO  WER.  3  7 

111.  A  fire-engine  pump  is  provided  with  a  nozzle, 
the  sectional  area  of  which  is  i  square  inch,  and  the 
water  is  projected  through  the  nozzle  with  an  average 
normal  velocity  of  1 30  feet  per  second.  Find  ( i ) 
the  number  of  cubic  feet  discharged  per  second  ;  (2) 
the  weight  of  water  discharged  per  minute ;  (3)  the 
kinetic  energy  of  each  pound  of  water  as  it  leaves 
the  nozzle  ;  (4)  the  horse-power  of  the  engine  re- 
quired to  drive  the  pump,  assuming  the  efficiency  to 
be  70  per  cent. 

112.  What  must  be  the  horse-power  of  a  pumping- 
engine  working  12  hours  per  day,  to  supply  50000 
persons  with  no  gallons  of  water  each  per  day,  sup- 
posing the  water  to  be  raised  to  the  mean  height  of 
190  feet,  the  efficiency  of  engine  and  pump  being  70 
per  cent  t 

113.  If  3.8  pounds  of  soft  coal  produces  one  horse- 
power, how  many  pounds  will  be  required  by  the  above 
pumping-engine  for  a  year's  supply  .? 

114.  Find  the  horse-power  necessary  to  pump  out 
the  Saint  Mary's  Falls  Canal  Lock,  Sault  Ste  Marie, 
in  24  hours,  the  length  of  the  lock  being  500  feet, 
width  80  feet,  and  depth  of  water  18  feet,  the  water 
being  delivered  at  a  height  of  42  feet  above  the  bot- 
tom of  the  lock. 

115.  There  were  6  000  cubic  feet  of  water  in  a 
mine  whose  depth  was  60  fathoms,  when  an  engine 
of  50  horse-power  began  to  operate  a  pump  ;  the  en- 
gine continued  to  work  5  hours  before  the  mine  was 


38  MECHANICS-PROBLEMS. 

cleared  of  water.  Supposing  one-fourth  of  the  work 
of  the  engine  to  have  been  wasted,  find  the  number 
of  cubic  feet  of  water  which  ran  into  the  mine  during 
the  5  hours. 

116.  A  nozzle  discharges  a  stream  i  inch  in  diame- 
ter with  a  velocity  of  8o  feet  per  second,  {a)  How 
much  kinetic  energy  is  possessed  by  the  amount  of 
water  which  flows  out  in  i  minute  t  {b)  If  this  en- 
ergy could  all  be  utilized  by  a  water-wheel,  what  would 
be  its  power } 

117.  Suppose  the  nozzle  referred  to  in  example  1 16 
to  drive  a  water-wheel  connected  with  a  pump  which 
lifts  water  20  feet.  If  the  efficiency  of  the  whole 
apparatus  is  0.48,  how  much  water  is  lifted  per 
minute } 

118.  The  mean  section  of ,  the  branch  of  the  First 
Level  Canal  at  the  headgates  of  No.  i  Mill,  Whiting 
Paper  Co.,  Holyoke,  Mass.,  is  yd)  feet  wide  by  14 
deep ;  from  this  canal  to  the  Second  Level  there  is  a 
fall  of  20  feet,  but  about  2  feet  is  lost  in  penstock  and 
tail-race.  The  turbines  that  are  driven  have  an  effi- 
ciency of  77  %.  Find  how  many  96-inch  Fourdrinier 
Paper  Machines  can  be  driven,  each  machine  requir- 
ing 100  horse-power. 

119.  What  is  the  horse-power  of  a  water-fall  of  1 8 
feet  when  the  stream  above  the  fall  passes  through  a 
section  of  6  square  feet  at  the  rate  of  2|-  miles  an 
hour. 


WORK— HORSE-POWER.  39 

120.  What  horse-power  is  involved  in  lowering  by 
2  feet  the  level  of  the  surface  of  a  lake  2  square  miles 
in  area  in  300  hours,  the  water  being  lifted  to  an 
average  height  of  5  feet  ? 

121.  Taking  the  average  power  of  a  man  as  ^^th 
of  a  horse-power,  and  the  efficiency  of  the  pump  used 
as  0.4,  in  what  time  will  10  men  empty  a  tank  of 
50  feet  X  30  feet  x  6  feet  filled  with  water,  the  lift 
being  an  average  height  of  30  feet  ? 

122.  A  shaft  560  feet  deep  and  5  feet  in  diameter 
is  full  of  water.  How  many  foot-pounds  of  work  are 
required  to  empty  it,  and  how  long  would  it  take  an 
engine  of  3|-  horse-power  to  do  the  work  1 

123.  Required  the  number  of  horse-power  to  raise 
2  200  cubic  feet  of  water  an  hour  from  a  mine  whose 
depth  is  63  fathoms. 

124.  What  weight  of  coal  will  an  engine  of  4  horse- 
power raise  in  one  hour  from  a  pit  whose  depth  is  200 
feet } 

125.  A  cut  is  being  made  on  a  4-inch  wrought-iron 
shaft  revolving  at  10  revolutions  per  minute;  the 
traverse  feed  is  0.3  inch  per  revolution;  the  pressure 
on  the  tool  is  found  to  be  435  pounds.  What  is  the 
horse-power  expended  at  the  tool  t  How  much  metal 
is  removed  per  hour  per  horse-power  when  the  depth 
of  cut  is  .06  inch,  the  breadth  .06  inch  (triangular 
section)  ? 


40  MECHANICS-PROBLEMS. 

126.  A  man  rides  a  bicycle  up  a  hill  whose  slope  is 
I  in  20  at  the  rate  of  4  miles  an  hour.  The  weight 
of  man  and  machine  is  iS/^  pounds.  What  work 
per  minute  is  he  doing  ? 

127.  At  the  top  of  the  hill  the  bicyclist  referred  to 
in  example  126  is  met  by  a  strong  head-wind,  and  he 
finds  that  he  has  to  work  twice  as  hard  to  keep  ths 
same  rate  of  4  miles  an  hour  on  the  level.  What 
force  is  the  wind  exerting  against  him  ? 

128.  A  bicyclist  works  at  the  rate  of  one-tenth  of  a 
horse-power,  and  goes  1 2  miles  an  hour  on  the  level. 
Prove  that  the  constant  resistance  of  the  road  is  3.125 
pounds. 

Prove  that  up  an  incline  of  i  vertical  to  50  horizon- 
tal the  speed  will  be  reduced  to  about  5.8  miles  per 
hour,  supposing  that  the  man  and  machine  together 
weigh  168  pounds. 

129.  A  man  rows  a  miles  per  hour  uniformly.  If  R 
pounds  be  the  resistance  of  the  water,  and  P  foot- 
pounds of  useful  work  are  done  at  each  stroke,  find 
the  number  of  strokes  made  per  minute. 

130.  A  train  runs  from  rest  down  an  incline  of  i  in 
100,  for  a  distance  of  i  mile  (no  engine  attached) ; 
it  then  runs  up  an  equal  grade  with  its  acquired 
velocity  for  a  distance  of  500  yards  before  stopping. 
Assuming  the  principle  of  work,  find  the  total  resist- 
ance, frictional  or  other,  in  pounds  per  ton,  which  has 
been  opposing  its  motion. 


WORK  —  HORSE-PO  WER.  4 1 

131.  In  the  Westinghouse  brake  tests  (Jan.,  1887), 
at  Weehawken,  a  passenger-train  moving  22  miles  an 
hour  on  a  down  grade  of  1%  was  stopped  in  91  feet. 
There  was  94%  of  the  train  braked.  Taking  the 
frictional  resistance  as  8  pounds  per  ton,  find  the  net 
brake  resistance  per  ton  and  the  grade  to  which  this 
is  equivalent. 

132.  The  rise  and  fall  of  the  tide  at  Boston,  Mass., 
is  about  9  feet.  If  the  in-coming  water  for  one 
square  mile  of  ocean  surface  could  be  stored,  and  its 
potential  energy  used  during  the  out-going  tide  with 
an  average  fall  of  4I  feet,  what  horse- power  would 
be  utilized } 

133.  A  six-inch  rapid-fire  gun  discharges  5  pro- 
jectiles per  minute,  each  of  weight  100  pounds,  with 
a  velocity  of  2  800  feet  per  second.  What  is  the 
horse-power  expended  t 

134.  A  500-volt  motor  drives  a  lo-ton  car  up  a  5 
per  cent  grade  at  a  speed  of  12  miles  per  hour:  75 
per  cent  of  the  work  of  the  motor  is  usefully  ex- 
pended. What  electric  current,  expressed  in  am- 
peres, will  be  required } 

.135.  The  resistance  offered  by  still  water  to  the 
passage  of  a  certain  steamer  at  10  knots  an  hour  is 
I  5  000  pounds.  A  portion  of  the  power  of  the  en- 
gines equal  to  12%  of  the  total  is  absorbed  in  the 
"  slip  "  (i.e.,  in  pushing  aside  and  backward  the  water 
acted  on  by  the  screw  or  paddle)  and  8  %  of  the 
total  is  absorbed  in  friction  of  machinery.  What 
must  be  the  total  horse-power  of  the  engines } 


4  2  ME  CHA  NJCS-PR  OBL  EMS. 

136.  The  United  States  warship  Columbia  has  a 
speed  of  23  knots,  with  an  indicated  horse-power  of 
22  000.     Find  the  resistance  offered  to  her  passage. 

137.  A  freight-car  weighing  20  000  pounds  requires 
a  net  pull  of  10  pounds  per  ton  to  overcome  frictional 
resistance.  If  "■  kicked  "  to  a  level  side  track  with 
velocity  of  10  miles  per  hour,  how  far  will  it  run  be- 
fore stopping.? 

138.  An  express  train  of  weight  250  tons  cover  40 
miles  in  40  minutes.  Taking  the  train  resistances  on 
a  level  track  to  be  20  pounds  per  ton  at  this  speed, 
find  the  horse-power  that  engine  must  develop. 

139.  The  speed  of  the  "  Exposition  Flyer  "  on  the 
Lake  Shore  and  Michigan  Southern  Railroad,  when 
running  at  its  maximum,  is  100  miles  per  hour.  At 
that  speed  what  pull  by  the  engine  would  represent 
one  horse-power  }  What  pull  when  running  at  50 
miles  an  hour  } 

140.  "  Up  to  the  highest  usual  speeds  of  commer- 
cial ships  we  may  assume  without  great  error  that, 
for  vessels  not  dissimilar  in  form  and  character  going 
at  the  usual  speeds,  the  indicated  horse-power  is  H  = 
D^V  ^-c  where  D  is  the  displacement  in  tons  and  V 
is  the  speed  in  knots  and  <:  is  a  constant,  which  for 
many  classes  of  vessel  maybe  taken  as  not  far  differ- 
ent from  240."  —  Perry's  Applied  MecJianics. 
What  is  the  indicated  horse-power  of  a  vessel  of  i  330 
tons,  moving  at  a  speed  of  12  knots,  if  it  obeys  the 
above  rule  '^. 


WORK  —  ENER  GV.  43 


E  NERGY 

141.  The  weight  of  a  ram  is  600  pounds,  and  at 
the  end  of  a  blow  it  has  a  velocity  of  40  feet  per 
second.     What  work  is  done  in  raising  it  ? 

142.  A  hoisting-engine  lifts  an  elevator  weighing 
I  ton  through  50  feet  when  it  attains  a  velocity  of 
4  feet  per  second.  If  the  steam  is  shut  off,  how 
much  higher  will  it  rise  ? 

143.  Find  the  horse-power  of  a  man  who  strikes 
25  blows  per  minute  on  an  anvil  with  a  hammer  of 
weight  14  pounds,  the  velocity  of  the  hammer  on 
striking  being  32  feet  per  second. 

The  height  from  which  hammer  would  have  to  fall  to 
acquire  the  same  velocity  would  be  found  from  the  fun- 
damental formula,  / — 7 


32  =  V64  X  /i 
.'.    /i  =  16  feet 
]VJow  work  =  force  X  distance. 

Or  work  may  be  found  directly  from  the  formula, 
work  =^  mv^, 

which  formula  is  easily  deduced  from  the  above. 

144.  Show  that  to  give  a  velocity  of  20  miles  an 
hour  to  a  train  requires  the  same  energy  as  to  lift  it 
vertically  through  a  height  of  13.4  feet. 

145.  What  is  the  kinetic  energy  of  a  cable  car  mov- 
ing at  6  miles  per  hour,  loaded  with  36  passengers, 
each    of    average    weight    1 54  pounds  ?     Weight  of 


44  MECHANICS-PROBLEMS. 

car,  2\  tons.  What  is  its  momentum  ?  If  stopped 
in  2  seconds,  what  is  the  average  force  ?  (The  space 
average  force  would  be  equivalent  to  a  constant  force 
sufficient  to  stop  the  car.) 

146.  What  is  the  kinetic  energy  of  an  electric  car 
weighing  2\  tons,  moving  at  lo  miles  an  hour,  and 
loaded  with  50  passengers,  each  of  average  weight 
150  pounds } 

147.  A  ball  weighing  5  ounces,  and  moving  at  i  000 
feet  per  second,  pierces  a  shield,  and  moves  on  with  a 
velocity  of  400  feet  per  second.  What  energy  is  lost 
in  piercing  the  shield  } 

148.  A  shot  of  I  000  pounds  moving  at  the  rate  of 
I  600  feet  per  second  strikes  a  fixed  target.  How  far 
will  the  shot  penetrate  the  target,  exerting  upon  it  an 
average  pressure  equal  to  a  weight  of   1 2  000  tons .? 

149.  A  bullet  weighing  i  ounce  leaves  the  mouth 
of  a  rifle  with  a  velocity  of  i  500  feet  per  second.  If 
the  barrel  be  4  feet  long,  calculate  the  mean  pressure 
of  the  powder,  neglecting  all  friction. 

Let  P  =  mean  pressure  of  powder  in  pounds 
Kinetic  energy  of  bullet  =  \  mv^ 

=  i  X  GV  -  32)  X  (1 500)^ 

Work  done  on  bullet  =  P  X  4 

.-.  P  X  4  =  { — z — )  X  (^  500/ 

V32  X  32/ 
I  500' 


4  X  32  X  32 
=  549^Vir  pounds  weight. 


WORK  —  ENEKG  Y. 


45 


150.  The  bullet  referred  to  in  example  149  pene- 
trates a  sand  bank  to  the  depth  of  3  feet.  What  is 
the  mean  pressure  exerted  by  the  sand,  and  how  long 
does  the  motion  continue  in  the  sand  I 

151.  A  bullet  leaves  the  barrel  of  a  gun  with  the 
velocity  of  i  000  feet  per  second  ;  supposing  it  to 
weigh  2  ounces,  find  the  work  stored  up  in  the  bullet 
and  the  height  from  which  it  must  fall  to  acquire  that 
velocity. 

152.  What  is  the  kinetic  energy  of  a  5-hundred 
weight  projectile  moving  with  a  velocity  of  2  000  feet 
per  second  t 

153.  A  half-ton  shot  is  discharged  from  an  81 -ton 
gun  with  a  velocity  of  i  620  feet  per  second.  With 
what  velocity  will  the  gun  recoil,  neglecting  the  mass 
of  the  powder  }  Will  the  gun  or  the  shot  be  able  to 
do  more  work  before  coming  to  rest,  and  in  what  pro- 
portion } 

154.  A  cannon  when  fired  recoils  with  a  velocity 
of  10  feet  per  second,  and  runs  up  a  platform  having 
an  incline  of  i  in  4.  Find  the  distance  (measured 
horizontally)  that  it  goes  before  coming  to  rest. 

155.  A  nail  2  inches  long  was  driven  into  a  block 
by  successive  blows  from  a  hammer  weighing  5.01 
pounds  ;  after  one  blow  it  was  found  that  the  head  of 
the  nail  projected  0.8  inches  above  the  surface  of  the 
block ;  the  hammer  was  then  raised  to  a  height  of 
1.5  feet  and  allowed  to  fall  upon  the  head  of  the  nail, 
which,  after  the  blow,  was  found  to  be  0.46  inches 


46  MECHANICS-PROBLEMS. 

above  the  surface.     Find  the  force  which  the  hammer 
exerted  upon  the  head  of  the  nail  at  this  blow. 

156.  A  hammer  weighing  i  pound  has  a  velocity 
of  20  feet  per  second  at  the  instant  it  strikes  the  head 
of  a  nail.  Find  the  force  which  the  hammer  exerts  on 
the  nail  if  it  is  driven  into  the  wood  -j^^th  of  an  inch. 

157.  What  is  the  energy  of  a  pendulum  bob  weigh- 
ing half  a  ton  and  swinging  past  its  equilibrium  posi- 
tion at  the  rate  of  i  foot  a  second  1 

158.  A  ball  weighing  8  pounds,  tied  to  one  end  of 
a  fine  thread  6  feet  long,  swings  backward  and  for- 
ward in  the  arc  of  a  semicircle.  What  is  the  kinetic 
energy,  and  what  is  its  velocity  as  it  passes  through 
the  lowest  point  of  the  semicircle .'' 

159.  The  weight  of  a  fly-wheel  is  8  000  pounds  and 
the  diameter,  20  feet;  diameter  of  axle,  14  inches  ; 
coefficient  of  friction,  0.2.  If  the  wheel  is  disconnected 
from  the  engine  when  making  27  revolutions  per 
minute,  find  how  many  revolutions  it  will  make  be- 
fore it  stops  (^  taken  =  32.2). 

„,    ,  1-1,1    8000   /20  X  27  X3iV 

Work  stored  in  wheel  =  - -z 

2     32.2     \  60  / 

Work  done  by  friction  in  x  revolutions 

St  X  14 

=  0.2  X  8  000  ~ ~  X  X 

12 

1  8  000  (20  X  27  X  2>^y-  o  3}  X  14,, 

-  ^ — -^ ^^  =  0.2x8  000  X  ^-^ X  X 

2  32.2  60  12 

^^5  X  81  X  3}  X  3 
32.2  X  7 
=  16.9  revolutions. 


IVOKK—  ENERG  Y.  4/ 

160.  A  small  heavy  body  weighing  20  pounds 
slides  down  a  rough  circular  arc  10  feet  in  radius 
whose  plane  is  vertical.  It  begins  to  move  from 
one  end  of  a  horizontal  diameter,  and  is  found  to 
reach  the  lowest  point  with  a  velocity  of  12  feet  a 
second.  How  many  foot-pounds  of  work  have  been 
done  against  friction  during  the  motion  ?  And  if  the 
same  proportionate  loss  of  energy  occurred  in  the 
next  portion  of  the  same  circle,  how  high  would  it 
ascend  ^. 

161.  Find  the  useful  work  done  each  second  by  a 
fire-engine  which  discharges  water  at  the  rate  of  500 
gallons  per  minute  with  a  velocity  of  72  feet  per 
second. 

162.  The  ordinary  fire-engine  when  in  full  opera- 
tion burns  soft  coal,  and  will  consume  in  an  hour  about 
60  pounds  per  fire-stream  of  250  gallons  per  minute. 
Therefore  at  the  Thanksgiving  fire  in  Boston,  in  1893, 
a  500-gallon  engine  that  was  running  48  hours  required 
how  many  pounds  of  coal  1 

163..  A  massive  slow-moving  fly-wheel,  mounted  on 
a' horizontal  axis  i  foot  in  diameter,  possesses  i  500 
foot-pounds  of  kinetic  energy,  which  is  used  to  raise 
a  weight  of  25  pounds  by  m.eans  of  a  rope  coiled  round 
the  axis.  Assuming  that  a  weight  of  5  pounds  is 
able  to  overcome  the  friction,  how  many  times  will 
the  wheel  revolve  before  it  comes  to  rest  1  How 
many  revolutions  in  the  opposite  direction  must  be 


48 


MECHANICS-PROBLEMS. 


made  before  the  original  energy  is  restored  to  the 
wheel  ? 

164.  A  fly-wheel  weighs  lo  ooo  pounds,  and  is  of 
such  a  size  that  the  matter  composing  it  may  be 
treated  as  if  concentrated  on  the  circumference  of  a 
circle  12  feet  in  radius.  What  is  its  kinetic  energy 
when  moving  at  the  rate  of  15  revolutions  a  minute  ? 
How  many  turns  would  it  make  before  coming  to 
rest,  if  the  steam  were  cut  off,  and  it  moved  against  a 
friction  of  400  pounds  exerted  on  the  circumference 
of  an  axle  i  foot  in  diameter?     (^=32.) 

165.  A  blacksmith's  helper  using  a  16-pound 
sledge  strikes  20  times  a  minute  and  with  a  velocity 
of  30  feet  per  second.      Find  his  rate  of  work. 

166.  A  body  weighing  200  pounds  is  moved  from  a 
state  of  rest,  and  is  found  subsequtnitly  to  be  moving 
at  the  rate  of  12  feet  a  second.  How  many  foot- 
pounds of  work  must  have  been  expended 
on  it  by  the  forces  exerting  motion  over 
and  above  those  e:-.pended  on  the  resis- 
tances } 

167.  In  a  jack-screw  the  pitch  of  the 
screw  is  i  inch,  the  lever  is  2  feet  long, 
and  the  force  applied  at  the  end  of  the 
lever  is  25  pounds.  Find  the  weight  that 
can  be  lifted,  friction  being  neglected. 


Fig.  10. 


168.    Determine  by  the  principle  of  work 
(neglecting  friction),  the  relation  between 


WORK—  ENERGY.  49 

the  effort  P  and  the  load  W  in  case  of  the  differential 
wheel-and-axle  of  Fig.  10. 

One  revolution, 

Work  of  P  =  P  X  2  ^TT 
Work  of  weight  =  4-W  X  2  /  x  tt  —  ^  W  X  2  ^tt 
P  X  2  TT^  =  I  Wtt  X  2  (/  —  r) 
P  X  2^  =  W(/-  r) 
P   _  r'-  r 
W  ~     2a    ' 

169.  A  body  is  suspended  by  an  elastic  string 
whose  unstretched  length  is  4  feet.  Under  a  pull  of 
10  pounds  the  string  stretches  to  a  length  of  5  feet. 
Required  the  work  done  on  the  body  by  the  tension 
of  the  string  while  its  length  changes  from  6  feet  to 
4  feet. 

170.  A  body  falls  down  the  whole  length  of  an  in- 
clined plane  on  which  the  coefficient  of  friction  is  0.2. 
The  height  of  the  plane  is  10  feet  and  the  base  30 
feet.  On  reaching  the  bottom  it  rolls  horizontally  on 
a  plane,  having  the  same  coefficient  of  friction.  Find 
how  far  it  will  roll. 

171.  Two  bodies  A  and  B,  weighing  50  pounds 
and  10  pounds  respectively,  are  connected  by  a 
thread ;  B  is  placed  on  a  smooth  table,  and  A  hangs 
over  the  edge.  When  A  has  fallen  10  feet,  what  is 
the  kinetic  energy  (or  accumulated  work)  of  the 
bodies   jointly,  and  what  of  them  severally  } 


50 


ME  CHA  NICS-PR  OBLE  MS. 


172. 


equilibrium. 


A  bead  whose  weight  is  W  is  free  to  slide 
on  a  smooth  circular  wire  in  a  vertical 
plane.  A  string  attached  to  the  bead 
passes  over  a  smooth  peg  at  the  high- 
est point  of  the  circle  and  sustains  a 
weight  P.  Determine  by  the  prin- 
ciple of   virtual  work   the  position   of 


FORCE.  51 


II.    FORCE 

FORCES    ACTING   AT   A    POINT 

173.  At  what  angle  must  the  forces  6  pounds 
and  8  pounds  be  incHned,  if  their  resultant  is  10 
pounds  ? 

Draw  the  parallelogram  of  forces,  one  side  being  6  units,  one  8, 
and  the  diagonal  between  them  10  units.  Required  the  angle  between 
the  6-  and  8-pound  forces.  Solve  by  trigonometry  or  geometry. 
Required  angle  =  90°. 

174.  What  is  the  resultant  of  forces  60  pounds 
and  80  pounds  acting  at  right  angles  to  each  other  ? 

175.  Two  men  pull  a  body  horizontally  by  means 
of  ropes.  One  exerts  a  force  of  28  pounds  directly 
-north,  the  other  a  force  of  42  pounds  in   direction  N. 

42°  E.     What    single  force  would  be    equivalent  to 
the  two } 

176.  Three  cords  are  knotted  together ;  one  of 
these  is  pulled  to  the  north  with  a  force  of  6  pounds, 
another  to  the  east  with  a  force  of  8  pounds.  With 
what  force  must  the  third  be  pulled  to  keep  the  whole 
at  rest } 

177.  Two  persons  lifting  a  body  exert  forces  of  44 
pounds  and  60  pounds  on  opposite  sides  of  the  ver- 


52 


ME  CHA  NICS-PR  OBL  EMS. 


Force  illustrated  by  two  fire  streams  being  delivered  by  the  pump  service 
of  the  large  cotton  mills  of  B.B.  «&  R.  Knight  at  Natick,  Rhode  Island.  One 
stream  is  being  held  by  men  in  correct  position,  the  other  by  men  who  have 
been  crowded  into  an  awkward  and  dangerous  position.  Pressur3  shown  on 
the  gauge  at  the  hydrant  was  75  pounds  per  square  inch. 


FORCES— AT  A    POINT.  53 

tical,  but  each  with  an  incHnation  of  28°.     What  single 
force  would  produce  the  same  effect  ? 

178.  A  force  of  50  units  acts  along  a  line  inclined 
at  an  angle  of  30°  to  the  horizon.  Find,  by  construe- 
tion  or  otherwise,  its  horizontal  and  vertical  com- 
ponents. 

179.  Explain  the  boatman's  meaning  when  he  says 
that  greater  force  is  developed  when  a  mule  hauls  a 
canal  boat  with  a  long  rope  than  with  a  short  one. 
Is  the  same  true  of  a  steam-tug  when  towing  a  four- 
master  1 

180.  Two  strings^  one  of  which  is  horizontal,  and 
the  other  inchned  to  the  vertical  at  an  angle  of  30°, 
support  a  weight  of  10  pounds.  Find  the  tension  in 
each  string. 

181.  Two  forces  of  20  pounds,  and  one  of  21  act 
at  a  point.  The  angle  between  the  first  and  second 
is  120°,  and  between  the  second  and  third,  30°. 
Find  the  resultant. 

182.  Forces  of  9  pounds,  12,  13,  and  26,  act  at  a 
point  so  that  the  angles  between  the  successive 
forces  are  equal.     Find  their  resultant. 

183.  A  weightless  rod,  3  feet  long,  is  supported 
horizontally,  one  end  being  hinged  to  a  vertical  wall, 
and  the  other  attached  by  a  string  to  a  point  4  feet 
above  the  hinge;  a  weight  of  120  pounds  is  hung 
from  the  end  supported  by  the  string.  Calculate  the 
tension  in  the  string,  and  the  pressure  along  the  rod. 


5  4  ^"^I^  CHA  A  'ICS-PROB  L  EAIS. 

184.  A  weight  of  loo  pounds  is  fixed  to  the  top  of 
a  weightless  rod  or  strut  5  feet  long  whose  lower  end 
rests  in  a  corner  between  a  floor  and  a  vertical 
wall,  while  its  upper  end  is  attached  to  the  wall  by  a 
horizontal  wire  4  feet  long.  Calculate  the  tension  in 
the  wire,  and  the  thrust  in  the  rod. 

185.  A  rod  AB  is  hinged  at  A  and  supported  in 
a  horizontal  position  by  a  string  BC  making  an 
angle  of  45°  with  the  rod  ;  the  rod  has  a  weight  of 
10  pounds  suspended  from  B.  Find  the  tension  in 
the  string  and  the  force  at  the  hinge.  (The  weight 
of  the  rod  can-  be  neglected.) 

186.  A  simple  triangular  truss  of  30  feet  span  and 
5  feet  depth  supports  a  load  of  4  tons 
at  the  apex.  Find  the  forces  acting  on 
rafters  and  tie  rod. 

187.    A  derrick    is    set    as  shown    in 
sketch,  the  load  being  8  tons.     Find  the 
Fig.  12.         stress  in  the  boom  and  the  tackle. 

188.  A  stiff -leg  steel  derrick,  with  mast  55  feet 
high,  boom  85  feet  long,  set  with  tackle  40  feet  long, 
as  shown  in  cut,  is  raising  two  boilers  of 
50  tons  weight.  Find  stresses  in  boom 
and  tackle.     (See  illustration  on  page  55.) 

189.  Find   the    stress    in    tackle    and 
compression    in   boom  of  towers  for  six- 
master  shown  on  page  24  when  bucket, 
weighing  with  its  load  2  tons,  is  set  in  position  showa 
by  Fig.  13. 


FORCES—AT  A   POINT. 


^fiTH  AR  Y 

^    OF  THE 


OF 


5 


Fig.  14. 

190.  A  balloon  capable  of  raising  a  weight  of  360 
pounds  is  held  to  the  ground  by  a  rope  which  makes 
an  angle  of  60°  with  the  horizon.  Determine  the 
tension  of  the  rope  and  the  horizontal  pressure  of  the 
wind  on  the  balloon. 

191.  A  uniform  beam  10  feet  long,  weighing  80 
pounds,  is  suspended  from  a  horizontal  ceiling  by  two 
strings  attached  at  its  ends,  and  at  points  16  feet 
apart  in  the  ceiling.      Find  the  tension  in  each  string. 

192.  A  boat  is  towed  along  a  canal  50  feet  wide, 
by  mules  on  both  banks  ;  the  length  of  each  rope 
from  its  point  of  attachment  to  the  bank  is  72  feet : 


56  MECHAKICS-PROBLEMS. 

the  boat  moves  straight  down  the  middle  of  the 
canal.  Find  the  total  effective  pull  in  that  direction, 
when  the  pull  on  each  rope  is  800  pounds. 

193.  A  boat  is  being  towed  by  a  rope  making  an 
angle  of  30°  with  the  boat's  length  ;  the  resultant 
pressure  of  the  water  and  rudder  is  inclined  at  60° 
to  the  length  of  the  boat,  and  the  tension  in  the  rope 
is  equal  to  the  weight  of  half  a  ton.  P'ind  the  re- 
sultant force  in  the  direction  of  the  boat's  length. 

194.  In  a  direct-acting  steam-engine  the  piston- 
pressure  is  22  500  pounds  ;  the  connecting-rod  makes 
a  maximum  angle  of  1 5°  with  the  line  of  action  of  the 
piston.      Find  the  pressure  on  the  guides. 

195.  A  man  weighing  160  pounds  sits  in  a  loop  at 
the  end  of  a  rope  10  feet  3  inches  long,  the  other 
end  being  fastened  to  a  point  above.  What  horizon- 
tal force  will  pull  him  2  feet  3  inches  from  the  verti- 
cal, and  what  will  then  be  the  pull  on  the  rope  '^. 

196.  A  man  weighing  160  pounds  sits  in  a  ham- 
mock suspended  by  ropes  which  are  inclined  at  30° 
and  45°  to  vertical  posts.     Find  the  pull  in  each  rope. 

197.    Two  equal  weights,  W,  are 

attached    to    the    extremities   of    a 

flexible  string    which    passes    over 

three    tacks  arranged  in  the  form 

Fig.  15-  Qf  ^j^    isosceles   triangle    with    the 

base  horizontal,  the  vertical  angle  at  the  upper  tack 

being  120°.     Find  the  pressure  on  each  tack. 


FORCES— AT  A    POINT.  57 

198.  A  rod  AB  5  feet  long,  without  weight,  is 
hung  from  a  point  C  by  two  strings,  which  are  at- 
tached to  its  ends  and  to  the  point ;  the  string  AC  is 
3  feet  long,  and  the  string  BC  2  feet  ;  a  weight  of 
2  pounds  is  hung  from  A  and  a  weight  of  3  pounds 
from  B.  Find  the  tension  of  the  strings  and  the 
condition  that  these  may  be  in  equilibrium. 

199.  A  weight  of  10  pounds  is  suspended  by  two 
strings,  7  and  24  inches  long,  the  other  ends  of 
which  are  fastened  to  the  extremities  of  a  rod  25 
inches  in  length.  Find  the  tension  of  the  strings 
when  the  weight  hangs  immediately  below  the  middle 
point  of  the  rod. 

200.  AB  is  a  wall,  and  C  a  fixed  point  at  a  given 
perpendicular  distance  from  it  ;  a  uniform  rod  of 
given  length  is  placed  on  C  with  one  end  against  AB. 
If  all  the  surfaces  are  smooth,  find  the  position  in 
which  the  rod  is  in  equilibrium. 

201.  AB  is  a  uniform  beam  weighing  300  pounds. 
The  end  A  rests  against   a  smooth  verti-  ^4 
cal  wall,  the  end  B  is  attached  to  a  rope 
CB.       Point    C    is    vertically    above    A,  X 
length  of  beam  is  4    feet,    rope    7   feet. 
Represent  the  forces  acting,  and  find  the 
pressure  against  the  wall  and  the  tension  ! 
in  the  rope.  .  Yi%.  16. 

202.  A  wagon  weighing  2  200  pounds  rests  on  a 
slope  of  inclination  30°.  What  are  the  equivalent 
forces  parallel  and  perpendicular  to  the  plane  ? 


58  MECHANICS-PROBLEMS. 

203.  AB  is  a  rod  that  can  turn  freely  round  one 
end  A ;  the  other  end  B  rests  against  a  smooth  in- 
cHned  jDlane.  In  what  direction  does  the  plane  react 
upon  the  rod  ?  Illustrate  your  answer  by  a  diagram 
showing  the  rod,  the  plane,  and  the  reaction. 

204.  A  wagon  weighing  2  tons  is  to  be  drawn  up 
a  smooth  road  which  rises  4  feet  vertically  in  a  dis- 
tance of  32  feet  horizontally  by  a  rope  parallel  to  the 
road.  What  must  the  pull  of  the  rope  exceed  in 
order  that  it  may  move  the  wagon .? 

205.  What  weight  can  be  drawn  up  a  smooth  plane 
rising  i  in  5  by  a  pull  of  260  pounds  {a)  when  the 
pull  is  parallel  with  the  plane  }  {b)  when  it  is  hori- 
zontal .'' 

206.  A  horse  is  attached  to  a  dump-car  by  a  chain, 
which  is  inclined  at  an  angle  of  45°  to  the  rails; 
the  force  exerted  by  the  horse  is  6^2  pounds.  What 
is  the  effective  force  along  the  rails  1 

207.  The  angle  of  inclination  of  a  smooth  inclined 
plane  is  45°  :  a  force  of  3  pounds  acts  horizontally, 
and  a  force  of  4  pounds  acts  parallel  to  the  plane. 
Find  the  weight  which  they  will  be  just  able  to 
support. 

208.  A  body  rests  on  a  plane  of  height  3  feet,  length 
5  feet.  If  the  body  weighs  14  pounds,  what  force  act- 
ting  along  the  plane  could  support  it,  and  what  would 
be  the  pressure  on  the  plane } 


FORCES  — AT  A    POINT.  59 

209.  A  number  of  loaded  trucks  each  containing 
one  ton,  standing  on  a  given  part  of  a  smooth  tram- 
way, where  the  inclination  is  30°,  support  an  equal 
number  of  empty  trucks  on  another  part,  where  the 
inclination  is  45°.      Find  the  weight  of  a  truck. 

210.  Two  planks  of  lengths  7  yards  and  6  yards 
rest  with  one  end  of  each  on  a  horizontal  plane,  the 
other  ends  in  contact  above  that  plane ;  two  weights 
are  supported  one  on  each  plank,  and  are  connected 
by  a  string  passing  over  a  pulley  at  the  junction  of 
the  planks  ;  the  weight  on  the  first  plank  is  21  pounds. 
What  is  the  weight  on  the  other,  friction  not  being 
considered  .'* 

211.  The  weight  of  a  wheel  with  its  load  is  2  tons, 
diameter  of  wheel  5  feet.  Find  the  least  horizontal 
force  necessary  to  pull  it  over  a  stone  4  inches  high. 
(When  the  wheel  begins  to  rise  three 
forces  are  acting  :  P,  W,  and  R  the 
reaction.     It  is  required  to  find  P.) 

212.  A  rectangular  box,  contain- 
ing  a  200-pound  ball,  stands  on  a  ^^^'  ^^' 
horizontal  table,  and  is  tilted  about  one  of  its  lower 
edges  through  an  angle  of  30.°     Find  the  pressure  be- 
tween the  ball  and  the  box. 

213.  An  iron  sphere  weighing  50  pounds  is  resting 
against  a  smooth  vertical  wall  and  a  smooth  plane 
which  is  inclined  60°  to  the  horizon.  Find  the  pres- 
sure on  the  wall  and  plane. 


6o  MECHANICS-PROBLEMS. 

214.  A  beam  weighing  400  pounds  rests  with  its 
ends  on  two  inclined  planes  whose  angles  of  inclina- 
tion to  the  horizontal  are  20°  and  30°.  Find  the 
pressures  on  the  planes. 

215.  A  thread  14  feet  long  is  fastened  to  two 
points  A  and  B  which  are  in  the  same  horizontal  line 
and  10  feet  apart  ;  a  weight  of  25  pounds  is  tied  to 
the  thread  at  a  point  P  so  chosen  that  AP  is  6  feet  — 
therefore  BP  is  8  feet  long.  The  weight  being  thus 
suspended,  find  by  means  of  construction  or  otherwise, 
what  are  the  tensions  of  the  parts  AP  and  BP  of  the 
thread. 

216.  AC  and  BC  are  two  threads  4  feet  and  5  feet 
long,  respectively,  fastened  to  fixed  points  A  and  B, 
which  are  in  the  same  horizontal  line  6  feet  apart ;  a 
weight  of  50  pounds  is  fastened  to  C.  Find,  by 
means  of  a  line  construction  drawn  to  scale,  the  pull 
it  causes  at  the  points  A  and  B.  Each  of  the  threads 
AC  and  BC  is,  of  course,  in  a  state  of  tension. 
What  are  the  forces  producing  the  tension } 

217.  A  boiler  weighing  3  000  pounds  is  supported 
by  tackles  from  the  fore  and  main  yards.  If  the 
tackles  make  angles  of  25°  and  35°  respectively  with 
the  vertical,  what  is  the  tension  of  each  .? 

218.  A  piece  of  wire  26  inches  long,  and  strong 
enough  to  support  directly  a  load  of  100  pounds, 
is  attached  to  two  points  24  inches  apart  in  the  same 
horizontal  line.     Find  the  maximum  load  that  can  be 


FORCES— AT  A    POINT.  6 1 

suspended  at  the  middle  of  the  piece  of  wire  without 
breaking  it. 

219.  A  picture  of  50  pounds  weight  hanging  ver- 
tically against  a  smooth  wall  is  supported  by  a  string 
passing  over  a  smooth  hook  ;  the  ends  of  the  string 
are  fastened  to  two  points  in  the  upper  rim  of  the 
frame,  which  are  equidistant  from  the  center  of  the 
rim,  and  the  angle  at  the  peg  is  60°.  Find  the  tension 
in  the  string. 

220.  A  weight  W  attached  by  two  connecting 
cords  of  lengths  a  and  b  to  two  fixed  points  A  and  B, 
and  separated  by  a  horizontal  interval  ^,  are  in  equilib- 
rium under  the  action  of  gravity.  Required  the 
stresses  P,and  Q  in  the  cords. 

221.  Two  equal  rods  AB  and  BC  are  loosely  jointed 
together  at  B.  C  and  A  rest  on  two  fixed  supports, 
in  the  same  horizontal  line,  and  are  connected  by  a 
cord  equal  in  length  to  AB.  If  a  weight  of  12  pounds 
be  suspended  from  B,  what  is  the  pressure  produced 
along  AB  and  BC,  and  the  tension  in  the  cord  } 

222.  A  new  device  for  unloading  cars  of  coal  is 
illustrated  and  described  luXh^  Journal  of  t/ie  Associa- 
tion of  Engineering  Societies  for  Octo- 
ber, 1 90 1.     The  loaded  car  is  taken  in  ^^;^°'"'" 

a  cradle,  and  the  coal  dumped  from 
the  car  into  a  bin  from  which  it  is 
distributed.  Total  weight  of  car,  coal 
and  cradle,  is  about  70  tons.  This 
weight  is  supported  by  two  sets  of  Y  '      Fig.  is. 


62  MECHANICS-PROBLEMS. 

braces  and  posts  about  as  shown  in  sketch.     Calcu- 
late the  stresses  acting  in  the  members, 

223.  A  tripod  whose  vertex  is  A,  and  whose  legs 
are  AB,  AC,  AD,  of  lengths  8  feet,  8.5,  and  9  re- 
spectively, sustains  a  load  of  2  tons.  The  ends 
B,  C,  D,  form  a  triangle  whose  sides  are  BC  7  feet> 
CD  6  feet,  BD  8  feet.  Find  by  graphical  construc- 
tion the  compressive  forces  in  each  leg. 

224.  Figs.  19-20  show  a 
pair  of  shears  erected  at 
Sparrow's  Point,  Md.,  for 
the  Maryland  Steel  Com- 
pany. The  two  front  legs 
are  hollow  steel  tubes  116 
feet    long.      They    are    45 

Fig.  19, 

feet  apart  at  the  bottom. 
The  back  leg  is  1 26  feet  long,  and  is  connected  to 
hydraulic  machines  for  operating  the  shears.  How 
much  are  the  forces  acting  in  these  legs  when  a 
Krupp  gun  weighing  122  tons  is  being  lifted.? 

225.  Each  leg  of  a  pair  of  shears  is  50  feet  long. 
They  are  spread  20  feet  at  the  foot.  The  back  stay 
is  75  feet  long.  Find  the  forces  acting  on  each 
member  when  lifting  a  load  of  20  tons  at  a  distance  of 
20  feet  from  the  foot  of  the  shear  legs,  neglecting  the 
weight  of  structure. 

226.  Shear  legs  each  50  feet  long,  30  feet  apart  on 
horizontal  ground,  meet  at  point  C,  which  is  45  feet 
vertically  above  the  ground  ;  stay  from  C  is  inclined 


FORCES— AT  A    POINT. 


63 


Fig.  20. 

at  40°  to  the  horizon  ;  a  load  of  10  tons  hangs  from 
C.     Find  the  force  in  each  leg  and  stay. 


227.  A  vertical  crane  post  is  10  feet  high,  jib  30 
feet  long,  stay  24  feet  long,  meeting  at  a  point  C. 
There  are  two  back  stays  making  angles  of  45°  with 
the  horizontal ;  they  are  in  planes  due  north  and  due 


FORCES  — AT  A    POINT. 


65 


west  fro:ii  the  post.  A  weight  of  5  tons  hangs  from 
C.  Fmd  the  forces  in  the  jib  and  stays —  ist,  when 
C  is  southeast  of  the  post ;  2d,  when  C  is  due  east ; 
3d,  when  C  is  due  south. 

228.  The  view  on  opposite  page  shows  one  of  the 
largest  dipper  dredges  ever  built,  the  "  Pan  American," 
constructed  at  Buffalo  in  1899  for  use  on  the  Great 
Lakes.  An  A-frame,  the  legs  of  which  are  57  feet 
long  and  40  feet  apart  at  the  bottom,  is  held  at  the 
apex  by  four  cables  which  are  100  feet  long.  The 
boom  is  53  feet  long  and  weighs  30  tons.  The 
handle,  which  weighs  about  4  tons,  is  60  feet  long, 
and  carries  on  its  end  a  dipper  weighing  16  tons, 
which  will  dredge  up  8}  cubic  yards,  or  about  12 
tons,  of  material  at  one  load. 

The  dipper  is  operated  by  a  wire  rope  passing  over 
a  pulley  on  the  outward  end  of  the  boom.  In  the 
position  represented  by  the  outline  sketch,  the  boom 
is  inclined  to  the 


water  surface  at 
an  angle  of  30°, 
the  dipper  is  car- 
rying the  full 
load,  and  the  han- 
dle is  in  a  hori- 
zontal position 
with  its  middle  point  supported  at  a  point  on  the 
boom  23  feet  from  the  foot  of  the  boom.  The  apex 
of  the  A-frame  is  vertically  above  the  foot  of  the 
boom.      Compute    the  forces  acting  in  the   100-foot 


Fig.  -21. 


66  MECHANICS-PROBLEMS. 

back-Stays,  (considering  them  to  be  one  rope  only,  in 
the  position  as  per  sketch)  in  the  legs  of  the  A-frame, 
in  the  boom,  and  in  the  wire  rope  which  raises  the 
dipper. 

229.  Draw  a  triangle  ABC  with  its  base  AB  hor- 
izontal, and  its  vertex  C  downwards  ;  let  AC  and  BC 
represent  threads  fastened  to  fixed  points  at  A  and 
B,  and  at  C  to  a  third  thread,  which  carries  a  given 
weight  W.  Given  the  angles  of  the  triangle  ABC, 
find  the  tensions  of  the  threads. 

230.  ABC  is  a  rigid  equilateral  triangle,  weight  not 
considered ;  the  vertex  B  is  fastened  by  a  hinge  to  a 
wall,  while  the  vertex  C  rests  against  the  wall  under 
B.  If  a  given  weight  is  hung  from  A,  find  the  reac- 
tions at  B  and  C.  What  are  the  magnitudes  and 
directions  of  the  forces  exerted  by  the  weight  on  the 
wall  at  B  and  C  } 

231.  ABCD  is  a  square ;  forces  of  i  pound,  6,  and 
9  act  in  directions  AB  and  AD,  respectively.  Find 
the  magnitude  of  their  resultant  correct  to  two  places 
of  decimals.     . 

232.  Draw  a  square  ABCD  and  its  diagonal  AC  ; 
two  forces  of  lo  units  act  from  A  to  B,  and  from  C 
to  D  respectively  forming  a  couple ;  a  third  force  of 
15  units  acts  from  C  to  A.  Find  their  resultant, 
and  show  in  a  diagram  how  it  acts. 


FORCES —  AT  A    POINT.  6/ 

233.  A,  B,  C,  D,  are  the  angular  points  of  a  square 
taken  in  order  ;  three  forces  act  on  a  particle  at  A, 
viz.  one  of  7  units  from  A  to  B,  a  second  of  10  units 
from  D  to  A,  and  a  third  of  5  V2  units  along  the 
diagonal  from  A  to  C.  Find,  by  construction  or 
otherwise,  the  resultant  of  these  three  forces. 

234.  Forces  P,  2P,  3P,  and  4P  act  along  the  sides 
of  a  square  A,  B,  C,  D,  taken  in  order.  Find  the 
magnitude,  direction,  and  line  of  action  of  the  result- 
ant. 


235.  A  sinker  is  attached  to  a  fishing-line  which  is 
then  thrown  into  running  water.  Show  by  means  of 
a  diagram  the  forces  which  act  on  the  sinker  so  as  to 
maintain  equilibrium. 

236.  A  uniform  rod  6  feet  long,  weighing  10  pounds, 
is  supported  by  a  smooth  pin  and  by  a  string  6  feet 
long  which  is  attached  to  the  rod  i  foot  from  one 
end  and  to  a  nail  vertically  above  the  pin,  4  feet  dis- 
tant. Show  by  construction  the  position  in  which 
the  rod  will  come  to  rest. 


237.  A  light  rod  AB  can  turn  freely  round  a  hinge 
at  A  ;  it  rests  in  an  inclined  position  against  a  smooth 
peg  near  the  end  B  ;  a  weight  is  hung  from  the  middle 
of  the  rod.  Show  in  a  diagram  the  forces  which 
keep  the  rod  at  rest,  and  name  them. 


I'iK 


68  MECFiANICS-PROBLEMS. 

238.  A  weight  W  on  a  plane 
inclined  30°  to  the  horizontal  is 
supported  as  shown  in  cut.  The 
angles  0  being  equal.  Find  the 
ratio  of  the  power  to  the  weight. 

239.  Discuss  the  action  of  the 
wind  in  propellinga  sailing-vessel. 

Let  AB  be  the  keel,  CD  the  sail.     Let  the  °v 

force  of  the  wind  be  represented  in  magnitude  p^jced_^ 
and  direction  by  EF.  The  component  GF 
of  EF,  perpendicular  to  the  sail,  is  the  effec- 
tive component  in  propelling  the  ship ;  the 
other  component  EG,  parallel  to  the  sail,  is 
useless ;  but  GF  drives  the  ship  forward  and 
sidewise.  The  component  GH  of  GF,  perpendicular  to  AB,  pro- 
duces side  motion,  or  leeway;  and  the  other  component  HF,  along 
the  keel,  produces  forward  motion,  or  headway. 

240.  A  sailing-boat  is  being 
driven  forward  by  a  force  of  300 
pounds  as  shown  in  Fig.  24. 
What  force  is  P  acting  in  direction 
of  motion  of  the  boat } 

241.  Discuss  the  action  of  the 
rudder  of  a  vessel  in  counteracting 

leeway.     Show  that  one  effect  of  the  action  of  the 
rudder  is  to  diminish  the  vessel's  motion. 

242.  A  thread  of  length  /  has  its  ends  fastened  to 
two  points  in  a  line  of  length  c,  and  inclined  to  the 
vertical  with  angle  0  ;  a  weight  W  hangs  on  the  thread 
by  means  of  a  smooth  hook.     Find  the  position  in 


FORCES.  — AT  A  POINT.  69 

which  the  weight  comes  to  rest  and  the  tension  in  the 
thread. 

243.  A  smooth  ring  weighing  40  pounds  sHdes 
along  a  cord  that  is  attached  to  two  fixed  points  in  a 
horizontal  line.  The  distance  between  the  points 
being  one-half  length  of  cord,  find  position  in  which 
weight  will  come  to  rest  and  the  tension  in  the  string 
near  the  points  of  attachment. 

244.  A  small  heavy  ring  A,  which  can  slide  upon 
a  smooth  vertical  hoop,  is  kept  in  a  given  position  by 
a  string  AB,  B  being  the  highest  point  of  the  hoop. 
Show  that  the  pressure  between  the  ring  and  the 
hoop  is  equal  to  the  weight  of  the  ring. 

245.  Draw  a  figure  showing  the  mechanical  con- 
ditions of  equilibrium  when  a  uniform  beam  rests  with 
one  extremity  against  a  smooth  vertical  wall,  and  the 
other  inside  a  smooth  hemispherical  bowl. 

246.  A  ball  8  inches  in  diameter,  weighing  100 
pounds,  rests  on  a  plane  inclined  30°  to  the  horizon, 
and  is  held  in  equilibrium  by  a  string  4  inches  long 
attached  to  a  sphere  and  to  an  inclined  plane.  Rep- 
resent the  forces  acting,  and  find  their  values. 

247.  A  uniform  sphere  rests  on  a  smooth  inclined 
plane,  and  is  held  by  a  horizontal  string.  To  what 
point  on  the  surface  of  the  sphere  must  the  string  be 
attached  }    Draw  a  figure  showing  the  forces  in  action, 


"JO  MECHANICS-PROBLEMS. 

assuming  that  the  weight  acts  through  the  center  of 
the  sphere. 

248.  A  particle  of  weight  W  is  supported  within  a 
smooth  hemispherical  bowl  by  a  string  of  given  length 
having  one  end  attached  to  a  point  in  the  rim.  State 
clearly  the  forces  which  keep  the  particle  in  equilib- 
rium ;  find  their  magnitudes  if  the  length  of  the 
string  and  the  radius  of  the  bowl  be  given  and  the 
rim  be  in  a  horizontal  plane. 

249.  A  uniform  bar  of  weight  20  pounds,  length 
1 2  feet,  rests  with  one  end  inside  a  smooth  hemispheri- 
cal bowl,  and  is  supported  by  the  edge  of  the  bowl  at 
a  point  distant  10  feet  from  the  above  end,  2  feet  of 
the  bar  being  outside  the  bowl.  Draw  the  forces 
producing  equilibrium,  and  find  their  values. 

250.  A  rod  of  weight  10  pounds  rests  in  a  smooth 
hemispherical  bowl,  which  is  fixed  with  its  rim  hori- 
zontal;  the  rod  is  12  feet  long,  and  2  feet  of  it  are 
outside  the  bowl.  The  inclination  of  the  rod  to  the 
horizon  is  30°.  Draw  a  figure  representing  the  re- 
actions of  the  bowl,  and  calculate  these  reactions. 

251.  The  platform  of  a  suspension  foot-br:dge  100 
feet  span,  10  feet  width,  supports  a  load,  including 
its  own  weight,  of  150  pounds  per  square  foot.  The 
two  suspension  cables  have  a  dip  of  20  feet.  Find 
the  force  acting  on  each  cable  close  to  the  tower,  and 
in  the  middle,  assuming  the  cable  to  har.g  in  a  para- 
bolic curve. 


FORCES— AT  A    POINT. 


252.  The  slopes  of  a  simple  triangular  roof-truss 
are  30°  and  45°,  and  the  span  is  50  feet.  The 
trusses  are  spaced  10  feet  apart,  and  the  weight  of 
the  roof  covering  and  snow  is  50 
pounds  per  square  foot.  Find  the 
tension  in  the  tie-rod. 

253.  In  a  roof  of  32  feet  span 
and  height  12  feet  the  trusses  are 
10  feet  apart,  and  the  members  EF,  GH,  come  to 
the  middle  points  of  the  rafters.  If  the  weight  of  the 
roof-covering  is  25  pounds  per  square  foot,  draw  the 
stress  diagram  and  scale  off  the  stressess. 


i 


Fig.  26. 


\  254.    A    highway    bridge 

of  span  50  feet,  breadth 
40  feet,  has  two  queen-post 
trusses  of  depth  8  feet  ; 
and  each  truss  is  divided  by  two  posts  into  three 
equal  parts.  The  bridge  is  designed  to  carry  a  load 
of  100  pounds  per  square  foot  of  floor  surface.  Find 
the  stresses  developed. 

255.    Find  the  stresses  in  a  king-post  truss  repre- 
sented in  the   figure.     Distance  | 
between     trusses    is     12     feet, 
height  of  truss   10  feet,  length 
of  span    40    feet,   uniform   load     T~  ~1 
of     200     pounds      per      linear               ^^^'  ^^* 
foot,  and  a  load  of   i  000  pounds  at  the  foot  of  the 
post. 


72  ME  CHA  NICS-  PROBLEMS. 


MOMENTS 


256.  The  length  of  a  bar  is  12  inches;  weights 
I  pound  and  2  pounds  are  attached  to  its  ends.  At 
what  point  must  the  bar  be  supported  to  effect  a 
balance } 

Draw  a  figure.  Let  distance  from  fulcrum  to  the  i -pound  weight 
be  X.  Suppose  that  weights  could  cause  one  complete  revolution 
around  fulcrum  ;  then 

Work  done  by  i-pound  weight  =Work  done  by  2-pound  weight 
Now,  work  done  by  i -pound  weights  force  x  distance 

=  I  X  2  TT  ji; 
Work  done  by  2-pound  weight  =  2  x  2  tt  {12  — x) 
:.   I  X  2  TT  jr=  2  X  2  TT  (12— x) 
and 

I  xx=2x(i2  — x) 

Or  in  words  :  The  weight  producing  motion  x  its  perpendicular 
lever  arm  =  the  weight  moved  x  its  perpendicular  lever  arm.  Thus 
observe  the  direct  relation  existing  between  the  principles  of  Work 
and  those  of  Moments. 

Definition.  —  The  Moment  of  a  force  about  a  point  is  the  prod- 
uct of  the  force  times  the  perpendicular  distance  from  the  point  to 
the  line  of  action  of  the  force  ;  or  briefly,  Moment  is  force  x  perpendic- 
ular.    Clockwise  motion  is  usually  taken  positive ;  opposite,  negative. 

From  the  above  equation,  x  would  be  found  equal  to  8  inches. 

257.  Two  weights  of  126  pounds  and  220  pounds 
respectively,  are  suspended  from  the  extremities  of  a 
straight  bar  26  inches  in  length.  Find  the  point  at 
which  their  resultant  will  intersect  the  bar. 

258-  A  rigid  rod  7  feet  long,  without  weight  rests 
on  a  fixed  point  2  feet  6  inches  from  one  end  ;  to  this 


FOR  CES  —  MOMENTS.  J  3 

end  a  weight  of  18  pounds  is  attached.  What 
weight  must  be  hung  from  the  other  end  so  that  the 
rod  may  be  horizontal  ? 

259.  A  hght  rod  of  length  3  yards  has  weights  of 
1 5  pounds  and  3  pounds  suspended  at  the  middle  and 
extremity  respectively ;  it  balances  on  a  fulcrum. 
Find  the  position  of  the  fulcrum,  and  the  pressure 
on  it. 

260.  A  stiff  pole  1 2  feet  long  sticks  out  horizon- 
tally from  a  vertical  wall.  It  would  break  if  a  weight 
of  28  pounds  were  hung  at  the  end.  How  far  out 
along  the  pole  may  a  boy  of  weight  1 1 2  pounds  ven- 
ture with  safety  1 

261.  The  length  of  an  oar  is  8  feet,  of  which  2 
feet  are  inside  the  rowlock  ;  a  man  exerts  a  pressure 
on  the  extremity  of  the  handle  of  100  pounds. 
What  is  the  pressure  on  the  rowlock,  and  resultant 
pressure  causing  the  boat  to  move  } 

262.  Find  the  propelling  force  on  an  eight-oared 
shell,  if  each  man  pulls  his  oar  with  a  force  of 
56  pounds,  and  the  length  of  the  oar  outside  the 
rowlock  is  three  times  the  length  inside  the  row- 
lock. 

263.  A  light  bar,  5  feet  long,  has  weights,  of  9 
pounds  and  5  pounds  respectively  suspended  from  its 


74  MECHANICS-PROBLEMS. 

extremities,    and   lo    pounds   from  its  middle    point. 
Find  the  point  on  which  it  will  balance. 

264.  A  lever  AB  of  the  first  order,  8  feet  long, 
has  the  fulcrum  2  feet  from  B,  a  weight  of  5  pounds 
is  hung  from  A,  and  one  of  1 7  pounds  from  B.  Put- 
ting the  weight  of  the  lever  itself  out  of  the  question, 
from  what  point  must  a  weight  of  2.5  pounds  be  hung 
to  keep  the  lever  horizontal } 

265.  A  weight  of  100  pounds  is  supported  by  a 
rope  which  passes  over  a  fixed  pulley  and  is  attached 
to  a  lever  at  a  point  2  feet  from  the  fulcrum  which  is 
at  the  end  ;  length  of  lever  is  1 2  feet.  What  weight 
must  be  suspended  at  the  other  end  to  keep  the  lever 
horizontal  1 

266.  Eight  sailors  raise  an  anchor,  of  weight  2  ()'^^ 
pounds,  by  pushing  on  the  spokes  of  a  capstan  which 
has  a  radius  of  14  inches.  If  they  all  push  at  equal 
distances  from  the  center  exerting  a  force  of  5  5  pounds 
each,  what  is  the  distance } 

Let  X  be  the  distance  from  the  axis. 
Take  moments  about  the  axis, 

8  X  56  X  X  =  2  688  X  14. 
.'.  :r  =  84  inches 
=  7  feet. 

267.  Is  there  any  reason  why  a  man  should  put 
his  shoulder  to  the  spoke  of  the  wheel  rather  than  to 
the  body  of  the  wagon  in  helping  it  up  hill } 


FORCES  —  MOMENTS.  J  5 

268.  A  rod  AB,  of  length  1 5  feet,  is  supported  by 
props  at  A  and  B  ;  a  weight  of  200  pounds  is  sus- 
pended from  the  rod  at  a  point  7  feet  from  A.  Find 
the  pressure  on  the  prop  at  A. 

269.  A  light  bar,  9  feet  long,  to  which  is  attached 
a  weight  of  150  pounds,  at  a  point  3  feet  from 
one  end,  is  borne  by  two  men.  Find  what  portion  of 
the  weight  is  borne  by  each  man,  when  the  bar  is 
horizontal. 

270.  A  light  rod,  16  inches  long,  rests  on  two  pegs 
9  inches  apart,  with  its  center  midway  between  them. 
The  greatest  weights,  which  can  be  suspended  sepa- 
rately from  the  two  ends  of  the  rod  without  disturb- 
ing the  equilibrium,  are  4  pounds  and  5  pounds  re- 
spectively. There  is  another  weight  fixed  to  the  rod. 
Find  that  weight  and  its  position. 

271.  A  light  rod  AB,  20  inches  long,  rests  upon 
two  pegs  whose  distance  apart  is  equal  to  half  the 
length  of  the  rod.  How  must  it  be  placed  so  that  the 
pressure  on  the  pegs  may  be  equal  when  weights  2W, 
3  W,  are  suspended  from  A,  B,  respectively .'' 

272.  The  horizontal  roadway  of  a  bridge  is  30  feet 
long  and  its  weight,  6  tons,  may  be  supposed  to  act 
at  its  middle  point,  and  it  rests  on  similar  supports 
at  its  ends.  What  pressure  is  borne  by  each  of  the 
supports  when  a  carriage  weighing  2  tons  is  one-third 
of  the  way  across  the  bridge  t 


^6  MECHANICS-PROBLEMS. 

273.  ''  We  have  a  vset  of  hay-scales,  and  some- 
times we  have  to  weigh  wagons  that  are  too  long  to 
go  on  them.  Can  we  get  the  correct  weight  by 
weighing  one  end  at  a  time  and  then  adding  the  two 
weights } " 

274.  A  rod,  i8  inches  long,  can  turn  about  one  of 
its  ends,  and  a  weight  of  5  pounds  is  fixed  to  a  point 
6  inches  from  the  fixed  end.  Find  the  force  which 
must  be  applied  at  the  other  end  to  preserve  equilib- 
rium. 

275.  A  straight  uniform  lever  weighing  10  pounds 
rests  on  a  fulcrum  one-third  of  its  length  from  one 
end ;  it  is  loaded  with  a  weight  of  4  pounds  at 
that  end.  Find  what  vertical  force  must  act  at  the 
other  end  to  keep  the  lever  at  rest. 

276.  A  weight  ot  56  pounds  is  attached  to  one  end 
of  a  uniform  bar  which  is  ten  feet  long,  and  weighs 
20  pounds ;  the  fulcrum  is  2  feet  from  the  end  to 
which  the  weight  is  attached.  What  weight  must  be 
applied  at  the  other  end  to  balance  } 

277.  AB  is  a  horizontal  uniform  bar  i|  feet  long,  and 
F  a  point  in  it  10  inches  from  A.  Suppose  that  AB 
is  a  lever  turning  on  a  fulcrum  under  Y,  and  carrying 
a  weight  of  40  pounds  at  B  ;  weight  of  lever,  4  pounds. 
If  it  is  kept  horizontal  by  a  fixed  pin  above  the  rod,  7 
inches  from  F  and  3  inches  from  A,  find  the  pressure 
on  the  fulcrum  and  on  the  fixed  pin. 


FORCES  —  MOMENTS.  J  / 

278.  A  lever,  i6  feet  long,  balances  about  a  point 
4  feet  from  one  end  ;  if  a  weight  of  1 20  pounds  be 
attached  to  the  other  end,  it  balances  about  a  point  6 
feet  from  that  end.      Find  the  weight  of  the  lever. 

279.  A  uniform  lever  is  18  inches  long,  and  each 
inch  in  length  weighs  i  ounce.  Find  the  place  of  the 
fulcrum  when  a  weight  of  27  ounces  at  one  end  of  the 
lever  balances  a  weight  of  9  ounces  at  the  other  end. 

280.  A  rod,  of  weight  4  pounds,  and  of  length  16 
feet,  balances  about  a  point  4  feet  from  one  end.  If  a 
weight  of  10  pounds  be  hung  2  feet  from  this  end, 
find  the  weiglit  that  must  be  hung  from  the  other 
extremity  that  the  rod  may  then  balance  about  its 
middle  point. 

281.  A  piece  of  shafting  10  feet  long,  and  weighing 
100  pounds,  rests  horizontally  on  two  horses  placed 
at  its  ends.     A  pulley  weighing      <  ,  j,,  > 

75  pounds  is  keyed  to  the  shaft      | :  c      ^^  ^^  > 

at  a  point  distant  3!  feet  from  1    ,4, 

^  100  lU.  75  lis. 

one    end.       How    many    pounds  Fig.  28. 

applied  at  the  other  end  by  a  man  lifting  vertically 
will  just  raise  it } 

100  pounds  the  weight  of  shaft  acts  downward  at  the  middle 
point;  75  pounds  the  weight  of  pulley  acts  downward  at  D,  3^  feet 
from  B.     Find  the  required  force  acting  upward. 

Take  moments  about  B, 

+  P  X  AB-ioo  X  CB  — 75  X  BD=o. 

.-.  P  X  10  =3  100  X  5  +  75  X  Z\' 
P  =  75  pounds. 


78 


MECHANICS-PROBLEMS. 


282.  Six  men  are  to  carry  an  iron  rail  60  feet  long 
and  weighing  90  pounds  per  yard  ;  each  man  sustains 
one-sixth  of  the  weight.  Two  men  are  to  lift  from 
one  end  and  the  other  four  by  means  of  a  cross-bar. 
Where  must  the  cross-bar  be  placed  t 

283.  A  block  of  stone  weighing  300  pounds  is  to 
be  removed  by  two  men  using  a  light  plank  6  feet 
long.  How  must  the  stone  be  placed  so  that  one  nan 
will  carry  two-thirds  of  the  weight  and  the  other  one- 
third  } 

284.  A  rod  2  feet  long,  with  a  weight  of  7  pounds 
at  its  middle  point,  is  placed  upon  two  nails,  A  and  B. 
AB  is  horizontal  and  7  inches  long.  Find  the  distances 
to  which  the  ends  of  the  rod  must  extend  beyond  the 
nails,  if  the  difference  of  the  pressures  on  the  nails 
be  5  pounds. 

285.  A  davit  is  supported  by  a  foot- 
step A  and  a  collar  B,  placed  5  feet 
apart.  A  boat  weighing  2  tons  is  about 
'^^  to  be  lowered,  and  is  hanging  4  feet 
horizontally  from  vertical  through  the 
foot-step  and  collar.  Determine  the 
forces  which  must  be  acting  at  A 
andB. 


286.  The  resistance  of  208  pounds 
found  in  example  80  pulls  out  of  ver- 
tical a  triangular  mass  of  rocks  by 
acting  at  P  as  represented  in  sketch. 
The  thickness  of  the  triangular  mass 


Fig.  30. 


FORCES— MOMENTS.  79 

being  i|  feet,  weight  of  stone  150  pounds  per  cubic 
foot,  and  voids  taking  one-third  of  space,  find  total 
weight  of  rocks,  and  height  that  center  of  gravity 
will  be  raised  if  P  acts  through  a  distance  of  30  feet. 

287.  Like  parallel  forces  of  10  and  20  units  act 
perpendicularly  to  AB  at  A  and  B  ;  a  force  of  1 5 
units  acts  from  A  to  B.  Find  the  resultant  of  the 
three  forces,  and  show  in  a  diagram  how  it  acts. 

288.  A  rod  is  acted  on  at  one  end  by  a  force  of  3 
downwards,  and  at  a  distance  of  two  feet  from  this 
end  by  a  force  of  5  upwards.  Where  must  a  force  of 
2  be  applied  to  keep  the  rod  at  rest .? 

289.  Three  parallel  forces  of  i  pound  each  act  on 
a  horizontal  bar.  The  right  hand  one  acts  vertically 
upwards,  the  two  others  vertically  downwards,  at  dis- 
tances 2  feet  and  3  feet  respectively,  from  the  first. 
Draw  their  resultant,  and  state  exactly  its  magnitude 
and  position. 

290.  A  rod  is  suspended  horizontally  on  two  points, 
A  and  B,  1 2  feet  apart ;  between  A  and  B  points 
C  and  D  are  taken,  such  that  AC  =  BD  =  3  feet ;  a 
weight  of  120  pounds  is  hung  at  C,  and  a  weight  of 
240  pounds  at  D  ;  the  weight  of  the  rod  is  neglected. 
Take  a  point  O,  midway  between  A  and  B,  and  find 
with  respect  to  O  the  algebraical  sum  of  the  moments 
of  the  forces  acting  on  the  rod  on  one  side  of  O. 

291.  A  horizontal  rod  without  weight,  6  feet  long, 
rests  on  two  supports  at  its  extremities  ;  a  weight  of 


8o  MECHANICS-PROBLEMS.  * 

672  pounds  is  suspended  from  the  rod  at  a  distance  of 
2\  feet  from  one  end.  Find  the  reaction  at  each 
point  of  support.  If  one  support  could  bear  a  pres- 
sure of  only  1 1 2  pounds,  what  is  the  greatest  distance 
from  the  otlier  support  at  which  the  weight  could  be 
suspended } 

292.  Three  equal  parallel  forces  act  at  the  corners 
of  an  equilateral  triangle.  Find  the  point  of  applica- 
tion of  their  resultant. 


293.  Find  the  center  of  the  three  parallel  forces  4 
pounds,  6,  and  8,  which  act  respectively  at  the  cor- 
ners of  an  equilateral  triangle. 

294.  P,  Q,  R,  are  parallel  forces  acting  in  the  same 
direction  at  the  angular  points  respectively  of  an 
equilateral  triangle  ABC.  If  P  =  2Q  =  3R,  find  the 
position  of  their  center ;  also  find  its  position  if  the 
direction  of  the  force  Q  is  reversed. 

295.  Show  that  if  two  forces  be  represented  in 
magnitude  and  direction  by  two  sides  of  a  triangle, 
taken  in  order,  the  sum  of  their  moments  about  every 
point  in  the  base  is  the  same. 

296.  Draw  a  square  whose  angular  points  in  order 
are  A,  B,  C,  D,  and  suppose  equal  forces  (P)  to  act 
from  D  to  A,  A  to  B,  and  B  to  C  respectively,  and  a 
fourth  force  (2P)  to  act  from  C  to  D.      Find  a  point 


FORCES— MOMENTS. 


8i 


such  that,  if  the   moments  of  the  forces  are  taken 
with  respect  to  it,  the  algebraic  sum  is  zero. 

297.    A  BCD   is   a  square,  the  length  of  each   side 
being  4  feet,  and  four  forces  act  as  follows  :  2  pounds 
from    D  to  A,  3  pounds  from  B  to  A, 
4  pounds  from  C  to   B,  and  5   pounds 
from  D  to  B.     Find  the  algebraical  sum    i 
of  the  moments  of  the  forces  about  C. 


The  forces  act  as  in  the  figure. 
Draw  CM  perpendicular  to  DB. 
Then,  CM  =  DM. 

.-.  CD2-CM2  +  MD2=  2CM2. 
CD 


Fig.  31. 


CM  = 


CM=: 


4    =  2.83  nearly. 


.•.  Algebraical  sum  of  the  moments  about  C 

=  -2XDC+3XCB  +  4Xo  —  5XCM 

=  -2X4  +  3X4  +  0  +  5  (2.83) 
=  —  8+  12  X  14.15 
—  —  10.15  units. 

298.  ABCD  is  a  square,  and  AC  is  a  diagonal : 
forces  P,  Q,  R,  act  along  parallel  lines  at  B,  C,  D,  re- 
spectively, Q  acts  in  the  direction  A  to  C,  P  in  same 
direction,  and  R  in  opposite  direction.  Find,  and 
show  in  a  diagram,  the  position  of  the  center,  when 
Q  =  5P  and  R  =  7P. 

299.  Draw  a  rectangle,  ABCD,  such  that  the  side 
AB  is  three-fourths  of  the  side  BC  ;  forces  of  3,  9, 
and  5  units  act  from  B  to  A,  B  to  C,  and  D  to  A  re- 
spectively.     Find  their   resultant  by  construction  or 


82 


MECHA  NICS-PROBLEMS. 


Otherwise,  and  show  in  your  diagram  exactly  how  it 
acts. 

300.  Prove  that,  if  parallel  forces  i,  2,  3,  4,  5,  6, 
are  situated  at  the  angles  of  a  regular  hexagon,  the 
distance  of  their  center  from  the  center  of  the  cir- 
cumscribing circle  is  two-sevenths  of  the  radius  of 
that  circle. 

301.  Six  forces,  represented  by  the  sides  of  a 
regular  hexagon  taken  in  order,  act  along  the  sides 
to  turn  the  hexagon  round  an  axis  perpendicular  to  its 
plane.  Show  that  the  moment  of  the  forces  is  the 
same  through  whatever  point  within  the  hexagon  the 
axis  passes. 

302.  A  triangular  table,  sides  8 
feet,  9  feet,  and  10  feet,  is  sup- 
ported by  legs  at  each  corner,  and 
350  pounds  is  placed  on  it  3  feet 
from  the  8-foot  side,  2  feet  from 
the    9-foot     side,    and     2.6     feet 

from  the     10-foot  side.     What  are  the  pressures  on 

the  legs } 

303.    A    triangular  shaped  platform  right-angled  at 

A,   with    side    AB    10     b  ^^^ 

feet  long,  side  AC  40    j 

feet     long,     is     loaded 

with     freight     at      50  ^^ 

pounds  per  square  foot  ^^2-  33. 

surface.     Find  the  load  carried  by  each  of  the  three 

corner-posts. 


FOR  CES  —  MOMENTS.  8  3 

O  is  center  of  gravity,  \  distance  from  base  to  vertex 
Area  =  40  x  ^  X  10 

=  200  square  feet 
Load  =  200  X  50  =  10  000. 

Moments  about  axis  AB, 

■\-  z  X  CA—  10  000  X  perp.  from  C.G.  to  axis  AB  =  o 

5  X  40  —  10  000  X  5  X  40  =  o 

2  X  40  =10  000  X  ^  X  40 

^=10000=333331. 

Moments  about  CA, 

y  y.  10  —  10  000  X  ^  X  10  =  o 
J  =  3  2>ZZ\' 
Moments  about  CB, 

X  X  perp.  to  BC  —  10  000  X  \  perp.  to  BC  =  o 

304.  A  floor  20  X  30  feet  is  supported  mainly  by  four 
posts,  one  at  each  corner.  There  is  a  load  of  20 
pounds  per  square  foot  uniformly  distributed,  and  at 
point  O,  5  feet  from  30-foot  side  and  7  feet  from  20- 
foot  .  side  there  is  a  metal  planer  weighing  5  tons. 
Find  the  load  on  each  post. 

305.  Weights  5,  6,  9,  and  7  respectively,  are  hung 
from  the  corners  of  a  horizontal  square,  27  inches  in 
a  side.  Find,  by  taking  moments  about  two  adjacent 
edges  of  the  square,  the  point  where  a  single  force 
must  be  applied  to  balance  the  effect  of  the  forces  at. 
the  corners. 

306.  Four  vertical  forces,  5,  7,  10  and  12  pounds^ 
act  at  the  corners  of  a  square  of  20-inch  sides.  Find 
resultant  and  its  point  of  application. 


84  MECHANICS-PROBLEMS. 

Let  ABCI)  be  the  square, 

Resultant  =  5  +  7  +  10+12 
=  34  pounds. 
To  find  its  point  of  application : 
Resultant  of  7  and  10  will  be  a  force 
of  17    pounds  acting   from    point    in   line 
CB  distant  -f-^  of  20  inches  from  B.     The 
resultant  of  5  and   12    will  be    17    pounds 


of  20  inches  from  A.  The  resultant  of 
these  two  resultants  will  be  a  force  of  17 
+  1 7  pounds,  34  pounds,  acting  at  a  point 
half  way  between  them  and  at  a  perpen- 
dicular distance  from  AB  of 
\oi  [j%  X  20  +  y\  X  20]  =  7^  inches. 

307.  A  uniform  beam,  weighing  400  pounds,  is 
suspended  by  means  of  two  chains  fastened  one  at 
each  end  of  the  beam.  When  the  beam  is  at  rest  it 
is  found  that  the  chains  make  angles  of  100°  and  1 1 5° 
with  the  beam.     Find  the  tension  in  the  chains. 

308.  What  is  the  resultant  of  a  couple  of  moment 
15,  and  a  force  3  ? 

309.  A  brakeman  sets  up  a  brake  on  a  freight 
car  by  pulling  50  pounds  with  one  hand  and  pushing 
50  pounds  with  the  other  ;  his  forces  act  tangentially 
to  the  brake  wheel,  the  diameter  of  which  is  i  J  feet. 
Another  time  he  produces  the  same  brake  resistance 
by  using  a  lever  in  handwheel  and  pulling  25  pounds. 
How  far  from  handwheel  must  his  hands  be  placed  t 

:  310.    When,  are  couples  said  to  be  like  and  wheji 
unlike  .''     When  will  two  unlike  couples   balance  .each 


FORCES— MOMENTS.  85 

Other?  (i)  If  a  system  of  forces  is  represented  in 
magnitude  and  position  by  the  sides  of  a  plane  poly- 
gon taken  in  order,  show  that  the  system  must  be 
equivalent  to  a  couple.  (2)  If  the  sides  of  a  parallelo- 
gram taken  in  order  represent  a  system  of  forces  act- 
ing upon  a  body,  express  the  moment  of  the  couple  to 
which  the  system  of  forces  is  equivalent. 

311.  Show  that  a  force  and  a  couple  in  one  plane 
may  be  reduced  to  a  single  force.  Given  in  position 
a  force  of  10  pounds,  and  a  couple  consisting  of  two 
forces  of  4  pounds  each,  at  a  distance  of  2  inches, 
acting  with  the  hands  of  a  clock,  draw  the  equivalent 
single  force. 

312.  The  length  of  the  side  of  a 
square  ABCD  is  12  inches.  Along 
the  sides  AB  and  CD  forces  of  10 
pounds  act,  and  along  AD,  CB  forces 
of  20  pounds.  Find  the  moment  of 
the  equivalent  couple. 

Moments  about  D, 

—  12X10+12  X2o  =  moment  of  equivalent-couple 
12  X  10  =  moment  of  equivalent-couple 

313.  Forces  P  and  Q  act  at  A,  and  are  completely 
represented  by  AB  and  AC,  sides  of  a  triangle 
ABC.  Find  a  third  force  R  such  that  the  three 
forces  together  may  be  equivalent  to  a  couple  whose 
moment  is  represented  by  half  the  area  of  the  triangle. 

314.  A  tradesman  has  a  balance  with  arms  of  un- 
equal length,  but  tries  to  be  fair  by  weighing  his  ma- 


S6  MECHANICS-PROBLEMS. 

terial  first  from  one  scale  pan,  then  from  the  other. 
Show  that  he  will  defraud  himself. 

315.  A  tradesman  uses  a  balance  with  arms  in 
ratio  of  5  to  6  ;  he  weighs  out  from  alternate  pans 
what  appears  to  be  30  pounds.  How  much  does  he 
gain  or  lose  .-* 

316.  The  beam  of  a  balance  is  6  feet  long,  and  it 
appears  correct  when  empty ;  a  certain  body  placed 
in  one  scale  weighs  120  pounds,  when  placed  in  the 
other,  121  pounds.  Show  that  the  fulcrum  must  be 
distant  about  ^^  of  an  inch  from  the  center  of  the 
beam. 

317.  The  weight  of  a  steelyard  is  1 2  pounds,  its 
movable  weight  is  3  pounds.  Find  the  distance 
between  successive  pound  graduations,  if  the  length 
of  the  short  arm  is  3  inches. 

318.  A  weight  of  247  pounds  is  attached  to  one 
end  of  a  horizontal  straight  lever,  which  is  22  inches 
long,  and  may  be  regarded  as  having  no  weight ; 
the  force  is  applied  at  the  other  end,  and  makes  an 
angle  of  27""  with  the  lever;  the  fulcrum  is  3  inches 
from  the  weight.  Find  the  magnitude  of  the  force 
when  it  just  balances  the  weight. 

319.  A  uniform  beam  rests  at  a 
given  inclination,  B,  with  one  end 
against  a  smooth  vertical  wall,  and 
the  other  end  on  smooth  horizontal 
ground :  it  is  held  from  slipping  by 
Fig.  36.  a  string  extending  horizontally  from 


FORCES— MOMENTS.  8/ 

the  foot  of  the  beam  to  the  foot  of  the  wall.  Find 
the  tension  in  the  string  and  the  pressures  at  the 
ground  and  wall. 

AB  is  the  beam,  AC  the  wall,  BC  the  string,  W  the  weight  of 
the  beam  acting  at  its  middle  point  G. 

There  are  three  forces  supporting  the  beam  :  vertical  reaction  P, 
horizontal  reaction  R,  and  tension  in  the  string  F. 

Take  moments  about  B,  the  point  of  intersection  of  two  of  the 
forces  —  their  lever  arms  would  be  zero. 

BC 
Rx  AC  =  W  x-:^. 

2 

Substitute  for  AC  its  value  BC  X  tan  d,  then 

W 


(i)R  = 


2  tan  e 


but  R  must  equal  F,  both  being  horizontal  resisting  forces  that  main- 
tain equilibrium;  likewise  P  and  W  must  be  equal. 

W 

.-.  (2)  F and 

^  '  2  lan  Q 

(3)  P  =  W 

320.  A  uniform  beam  rests  with  a  smooth  end 
against  the  junction  of  the  horizontal  ground  and  a 
vertical  wall ;  it  is  supported  by  a  string  fastened  to 
the  other  end  of  the  beam  and  to  a  staple  in  the  ver- 
tical wall.  Find  the  tension  of  the  string,  and  show 
that  it  will  be  half  the  weight  of  the  beam  if  the 
length  of  the  string  be  equal  to  the  height  of  the 
staple  above  the  ground. 

321.  A  uniform  rod  8  feet  long,  weighing  i8 
pounds,  is  fastened  at  one  end  to  a  vertical  wall  by  a 
smooth  hinge,  and  is  free  to  move  in  a  vertical  plane 
perpendicular  to  the  wall.  It  is  kept  horizontal  by  a 
string  lo  feet  long,  attached  to  its  free  end  and  to  a 


88       •  MECHANICS-PROBLEMS. 

point  in  the  wall.      Find  the  tension  in  the  string,  and 
the  pressure  on  the  hinge. 

322.  A  uniform  beam,  1 2  feet  in  length,  rests  with 
one  end  against  the  base  of  a  wall  which  is  20  feet 
high.  If  the  beam  be  held  by  a  rope  13  feet  long, 
attached  to  the  top  of  the  beam  and  to  the  summit  of 
the  wall,  find  the  tension  of  the  rope,  neglecting  its 
weight,  and  assuming  the  weight  of  the  beam  to  be 
lOO  pounds. 

323.  A  foot-bridge  with  1 8  feet  span,  6  feet  breadth 

\  has    two     king-post    trusses, 

one  on  each  side,  3  feet  deep. 

The    bridge     is    loaded    with 

^^^'  37-  people     which      makes      100 

pounds  per  square  foot  of  floor  surface.     Find    the 

stress  in  the  post. 

324.  A  beam  AB  rests  on  the  smooth  ground  at 
A  and  on  a  smooth  inclined  plane  at  B  ;  a  string  is 
fastened  at  B  and,  passing  over  a  smooth  peg  at  the 
top  of  the  plane,  supports  a  weight  P.  If  W  is  the 
weight  of  the  beam,  and  a  the  inclination  of  the  plane, 
find  P  and  the  reactions  on  the  rod. 

Draw  the  figure. 

The  weight  W  acts  at  the  middle  point  C.  The  reaction  of  the 
ground  at  A  is  R,  upwards. 

The  reaction  of  the  plane  at  B  is  Ri,  perpendicular  to  the  plane. 

Let  the  angle  BAD  —  B. 

The  tension  of  the  string  at  B  =  tension  of  the  string  throughout 
=  P. 

There  are  four  forces  acting  on  the  beam,  W,  R,  Ri,  P. 
.Resolve  vertically  and  horizontally. 


FOR  C  'ES  —  MOMENTS.  8  9 

325.  A  pole  12  feet  long,  weighing  25  pounds, 
rests  with  one  end  against  the  foot  of  a  wall,  and 
from  a  point  2  feet  from  the  other  end  a  cord  runs 
horizontally  to  a  point  in  the  wall  8  feet  from  the 
ground.  Find  the  tension  of  the  cord  and  the  pres- 
sure of  the  lower  end  of  the  pole. 

326.  A  light  smooth  stick  3  feet  long  is  loaded  at 
one  end  with  8  ounces  of  lead  ;  the  other  end  rests 
against  a  smooth  vertical  wall,  and  across  a  nail  which 
is  I  foot  from  the  wall.  Find  the  position  of  equi- 
librium and  the  pressure  on  the  nail  and  on  the  wall. 

327.  A  trapezoidal  wall  has  a  vertical  back  and  a 
sloping  front  face  ;  width  of  base,  10  feet ;  width  of 
top,  7  feet ;  height,  30  feet.  What  horizontal  force 
must  be  applied  at  a  point  20  feet  from  the  top  in 
order  to  overturn  it }  Thickness  of  wall,  i  foot ; 
weight  of  masonry  in  wall,  1 30  pounds  per  cubic  foot. 

328.  Six  men  using  a  rope  50  feet  long  were  just 
able  to  pull  over  a  chimney  75  feet  high.  How  far 
up  from  the  bottom  of  the  chimney  was  it  advisable 
to  attach  the  rope  } 

329.  If  150000  pounds  is  the  thrust  along  the 
connecting  rod  of  the  engine,  in  example  86,  2|  feet 
the  crank  radius,  and  the  connecting-rod  is  inclined 
to  the  crank  axis  at  150°,  show  that  the  moment  of 
the  thrust  about  the  crank-pin  is  one-half  the  greatest 
possible  moment. 

330.  A  trap-door  of  uniform  thickness,  5  feet  long 
and   3   feet  wide,   and  weighing  5  hundred  weight,  is 


go 


ME  CIIA  NICS-PK  OBL  EMS. 


held  open  at  angle  of  35*^  with  the  horizontal  by 
means  of  a  chain.  One  end  of  chain  is  hooked  at 
middle  of  top  edge  of  door,  and  the  other  is  fastened 
at  wall  4  feet  above  hinges.  Find  the  force  in  the 
chain  and  the  force  at  each  hinge. 

331.    The  sketch  represents  a  coal  wagon  weighing 
r^— -___  with  its  load  4|  tons.     How 

y  ~~~~~--~-,   many    pounds    applied   at 

^    \r~~^^-22^   ^-^         \    P  by  usual  methods  of  hand 
\       P/r^^^J^^^B  power   will    just    lift    the 
E  D         wagon  when   in    the   posi- 

tion shown  in  the  sketch  } 
AE   is  a  rod    in  tension.     CD    is  a  connecting-bar. 
Divide  the  problem  into  three  parts : 
{a)    Draw  the  forces  acting. 

{b )    Find  horizontal  distance  from  C  to  the  verti- 
cal through  the  center  of  gravity. 

( e)    Find  force  to  apply  at  C  parallel  to  P ;  then 
find  P. 


CENTER     OF     GRAVITY 

332.  A  rod  of  uniform  section  and  density,  weigh- 
ing 3  pounds,  rests  on  two  points,  one  under  each 
end  ;  a  movable  weight  of  4  pounds  is  placed  on 
the  rod.  Where  must  it  be  placed  so  that  one  of  the 
points  may  sustain  a  pressure  of  3  pounds,  and  the 
other  a  pressure  of  4  pounds  ? 


FORCES— CENTER    OF   GRAVITY.  9 1 

333.  Two  rods  of  uniform  density 
weighing  2  pounds  and  3  pounds  re- 
spectively are  put  together  so  that  the 
3-pound  one  stands  on  the  middle  of 
the  other.  Find  the  center  of  gravity  of  a  — 
the  whole.  ^'^'  ''' 

Take  moments  about  AB, 

+  3  X  i^—  5X^  =  0 
X  =  j\  of  /. 

334.  A  curtain  rod  5  feet. long,  weighing  4  pounds, 
has  four  rings  on  it  each  weighing  i  pound.  The 
two  end  rings  are  4  feet  apart,  the  two  middle 
rings  2  feet  apart,  and  one  ring  is  distant  6  inches 
from  the  middle  of  the  rod.  Find  their  center  of 
gravity. 

335.  Three  particles  of  masses,  3  pounds,  7  pounds, 
and  10  pounds,  are  respectively  5  feet  above,  6  feet 
above,  and  1 2  feet  below  a  horizontal  line.  What  is 
the  position  of  their  center  of  gravity  with  reference 
to  the  horizontal  line  ? 

336.  A  rod  ABC,  16  inches  long,  rests  in  a  horizon- 
tal position  upon  two  supports  at  A  and  B  one  foot 
apart,  and  it  is  found  that  the  least  upward  or  down- 
ward forces  applied  at  C  which  would  move  the  rod 
are  4  ounces  and  5  ounces  respectively.  Find  the 
weight  of  the  rod  and  the  position  of  its  center  of 
gravity. 

337.  A  straight  line  AB  represents  a  rod  10  feet 
long,  supported  horizontally  by  two  points,  one  under 


92  MECHANICS-PROBLEMS. 

each  end  ;  C  is  a  point  in  AB,  3  feet  from  A.  What 
pressure  is  produced  on  the  points  A  and  B  by  a 
weight  of  30  pounds  hung  at  C  ?  What  additional 
pressure  is  exerted  on  the  points  of  support  if  the 
rod  is  of  uniform  density  and  weighs  20  pounds? 

■   338.    A  thin  plate  of    metal  is  in  the  shape  of  a 
E  square  and  equilateral  triangle,   having  one 

side  common  ;  the  side  of  the  square  is  12 
^^"  ,^  inches  long.     Find  the  center  of  gravity  of 
the  plate. 

Let  Gi  be  the  center  of  gravity  of  the  triangle,  G2 
of  the  square,  G  of  the  whole  plate. 
From  symmetry  EG^  GG,0  will  b2  a  straight  line  bisecting  the 
plate,  and 

OG2  =  6  inches 
OGi  =15.5  inches 
Let  zv  =  weight  of  metal  per  square  inch 

Area  of  triangle  =  \  X  12  xV^^^—  6^ 
=  62.4  square  inches 
Weight  =  62.4  pounds  X  w  pounds 
Area  of  square  =144  square  inches 
Weight  =  144  X  ^<:^  pounds 
Take  moments  about  the  axis  AB, 
Weight  of  triangle  XOG1  + weight  of  square  xOGo— total 
weight  X  OG  =  o 
62.4Z£/X  i5.5  +  i44Z£/x6  — (62.4Z£/+i447e/)  X  OG  =  o 
.-.  OG  =  8.86  inches. 

339.    ABC  is  a  triangle  with  a  right  angle  at  A. 
AB  =  3  inches;  AC  =  4  inches;  weights  of  2  ounces, 


FORCES— CENTER   OF  GRAVITY. 


93 


3  and  4,  are  placed  at  A,  B,  and  C.     Find  the  position 
of  their  center  of  gravity. 

340.  A  uniform  triangle  ABC  of  weight  W,  and  ly- 
ing on  a  horizontal  table,  is  just  raised  by  a  vertical 
force  applied  at  A.  Find  the  magnitude  of  this  force, 
and  that  of  the  resultant  pressure  between  the  base 
BC  and  the  table. 

341.  A  uniform  circular  disk  has  a  circular  hole 
punched  out  of  it,  extending  from  the  circumference 
half  way  to  the  center.  Find  the  center  of  gravity 
of  the  remainder. 


342.  A  box,  including  its  cover,  is  made  of  six  equal 
square  boards ;  where  is  its  center  of  gravity  when 
its  lid  is  turned  back  through  an  angle  of  1 80°  ) 

343.  ABCD  is  a 

thin       rectangular 
plate    weighing    50  e^ 
pounds,    AB    is    10  n; 
feet,  BC  2  feet ;  the  .  i 
plate    is    suspended 
by  the  middle  point 
of   its    upper    edge  5' 
AB,    and    then,    of  ^^^^  ^'• 

course,  AB  is  horizontal,  but  if  a  weight  of  5  pounds 
is  placed  at  A,  AB  will  become  inclined  to  the  hori- 
zon. Show  how  to  find  the  angle  of  inclination 
either  by  calculation  or  by  construction. 


94 


ME  CHA  NICS-FR  OBLEMS. 


Moments  about  P, 

PGi  X  50  =  PN  X  5 

PGj  =  MGi  X  sin  Q 

=  I  X  sin  ^. 

From  the  above  equations, 

PN  =  EM  =  10  X  sin  0 

EA 

EM 

EA  =  MA  X  sin  0 

=  5  X  sin  ^ 

.        5  X  sin  ^ 

/.   tan  Q  =  ^ ^-. 

10  X  sm  ^ 


tan^ 


0  =  tan-i  i^ 

344.  A  circular  disk,  8  inches  in  diameter,  has  a 
hole  2  inches  in  diameter  punched  out  of  it,  the  center 
of  the  hole  being  3  inches  from  the  circumference 
of  the  disk.  Find  the  center  of  gravity  of  the  remain- 
ing portion. 


345.  Find  the  centers 
of  area  of  the  above  sec- 
tions of  uniform  plate 
metals. 


Fig.  42. 


346.  Into  a  hollow  cylindrical  vessel  1 1  inches 
high  and  w^eighing  10  pounds,  the  center  of  gravity 
of  which  is  5  inches  from  the  base,  a  uniform 
solid  cylinder  6  inches  long  and  weighing  20  pounds 
is  just  fitted.     Find  the  common  center  of  gravity. 


FORCES— CENTER    OF  GRAVITY.  95 

Gi  center  of  gravity  of  hollow  cylinder 
G2  center  of  gravity  of  solid  cylinder. 
Moments  about  AB, 

4-10x5  +  20X3  —  30X^  =  0 
+  50  +  60  —  30  a:  =  o 
30:1:=  no 
^  =  3|  inches.  "         Fig-  43. 

347.  Give  examples  of  stable  and  unstable  equilib- 
rium. A  cone  and  a  hemisphere  of  the  same  material 
are  cemented  together  at  the  common  circular  base. 
If  they  are  on  a  horizontal  plane,  and  the  hemisphere 
in  contact  with  the  plane,  find  the  height  of  the  cone 
in  order  that  the  equilibrium  may  be  neutral.  (The 
center  of  gravity  of  a  hemisphere  divides  a  radius  in  the 
ratio  of  3  to  5.) 

348.  A  thread  9  feet  long  has  its  ends  fastened  to 
the  ends  of  a  rod  6  feet  long ;  the  rod  is  supported 
in  such  a  manner  as  to  be  capable  of  turning  freely 
round  a  point  2  feet  from  one  end  ;  a  weight  is  placed 
on  the  thread,  like  a  bead  on  a  string.  Find  the 
position  in  which  the  rod  will  come  to  rest,  it  being 
supposed  that  the  rod  is  without  weight,  and  that  there 
is  no  friction  between  the  weight  and  the  thread. 

349.  A  circular  disk  weighs  9  ounces  ;  a  thin 
straight  wire  as  long  as  the  radius  of  the  circle  weighs 
7  ounces  ;  if  the  wire  is  placed  on  the  disk  so  as  to  be 
a  chord  of  the  circle,  the  center  of  gravity  of  the 
whole  will  be  at  a  distance  from  the  center  of  the 
circle  equal  to  some  fractional  part  of  the  radius. 
Find  that  fraction  by  construction  or  calculation. 


96  MECHANICS-PROBLEMS, 

350.  A  cone  and  a  hemisphere  are  on  the  same 
base.  What  height  must  the  cone  be  in  order  that  the 
center  of  gravity  of  the  whole  solid  shall  be  at  the 
center  of  the  common  base  ? 

r  —  radius  common  base. 
h  —  height  of  cone. 


FRICTION 

The  coefficients  of  friction  for  various  pairs  of  sub- 
stances have  been  found  experimentally  by  Morin  ; 
these  results  however  can  be  used  only  for  approxi- 
mate computation  ;  actual  trial  should  be  made  for 
specific  cases. 

Oak  on  oak,  fibers  parallel  to  direction  of  motion      .     0.48 

perpendicular 0.34 

endwise 0.19 

Metals  on  oak  dry,  fibers  parallel     ....      0.5    to  0.6 

Metals  on  metals,  dry 0.15  to  0.2 

Smooth  surfaces  with  unguents  well  greased      .     .     .     0.05 

351.  What  push  would  be  required  to  move  a  stone 
of  weight  3  pounds  along  ice,  if  the  coefficient  of 
friction  is  o.  i  1 

R  352.    A  weight  of  56  pounds  is  moved 

F^— ^->8ft,  along  a  horizontal  table  by  a  force  of  8 

I        56  14,.        1  pounds.      How    much  is  the  coefficient 

^«-  ^^'  of  friction  .? 

The  pull  of  8  pounds  is  required  to  overcome  friction,  and  is 
equal  to  the  friction. 

Friction  =  coefficient  X  Reaction  (perpendicular  to  plane  of  table). 


FORCES—  FRICTION.  97 

F  =  /x  X  R 

=  /A  X  56  pounds 
8  = /a  X  56 

353.  A  block  of  wood  weighing  i  pound  is  just 
dragged  along  a  horizontal  table  by  a  force  of  i 
pound.  What  is  the  direction  of  the  resultant  re- 
action } 

354.  What  is  the  angle  of  friction  or  limiting  angle 
of  resistance  t  When  a  body  urged  against  a  rough 
fixed  plane  by  certain  forces  is  at  rest,  to  what  extent 
is  the  direction  of  the  reaction  of  the  plane  against 
it  known  1 

355.  A  block  of  stone  is  dragged  along  the  ground 
by  a  horse  exerting  a  force  of  224  pounds.  If  /x  =  0.6, 
what  is  the  weight  of  the  block } 

356.  A  weight  of  500  pounds  is  placed  on  a  table, 
and  can  hardly  be  slid  by  a  horizontal  pull  of  155 
pounds.  Find  the  coefficient  of  friction,  and  the 
number  of  degrees  in  the  angle  of  friction  by  measur- 
ing from  a  drawing  made  to  a  scale. 

357.  A  stone  just  slides  down  a  hill  of  inclination 
30°.     What  is  the  coefficient  of  friction  } 

358.  A  block  rests  on  a  plane  which  is  tilted  till 
the  block  commences  to  slide.  The  inclination  is 
found  to  be  8.4  inches  at  starting,  and  afterwards  6.3 
inches  on  a  horizontal  length  of  2  feet.      Find  the  co- 


98  ME  CHA  NJCS-PROBLEMS. 

efficient  of  friction  when  the  block  starts  to  slide,  and 
after  it  has  started. 

359.  A  horse  draws  a  load  weighing  2  000  pounds 
up  a  grade  of  i  in  20 ;  the  resistance  on  the  level  is 
100  pounds  per  ton.  Find  the  pull  on  the  traces 
when  they  are  parallel  with  the  incline. 

360.  How  much  work  has  a  man,  weighing  224 
pounds,  done  in  walking  twenty  miles  up  a  slope  of  i 
vertical  to  40  horizontal }  What  force  could  drag  a 
dead  load  of  the  same  weight  up  the  same  hill  {a)  if 
friction  be  negligible,  {b)  if  friction  be  \  of  the 
weight  ? 

361.  Three  artillerymen  drag  a  gun  weighing 
I  700  pounds  up  a  hill  rising  2 
vertically  in  17  horizontally.  Sup- 
pose the  resistance  to  the  wheels 
going    up  the  hill    be    16    pounds 

per  hundred  weight,  what  pull  parallel  to  the  hill  must 
each  exert  to  move  it } 

When  the  gun  is  about  to  move  forward  the  pull  P  will  be  acting 
up  the  plane,  and  parallel  to  it ;  the  friction  F  down  the  plane,  hold- 
ing back;  the  force  R  perpendicular  to  incHned  plane,  partly  sup- 
porting the  gun,  and  W  the  weight  of  the  gun  acting  vertically  down- 
ward. Weight  of  gun  is  given —  i  700  pounds.  Resolve  into  com- 
ponents perpendicular  and  parallel  to  the  plane.  The  perpendicular 
component  will  be  the  supporting  force  of  the  plane  —  its  reaction 
R ;  the  parallel  component  will  be  the  part  of  the  pull  P  required  by 
weight  of  the  gun. 

362.  Find  the  force  which,  acting  in  a  given  direc- 
tion, will  just  support  a  body  of  given  weight  on  a 


FORGES  — FRICTION.  99 

rough  inclined  plane.  The  height  is  to  the  base  of  the 
plane  as  3  to  4,  and  it  is  found  that  the  body  is  just 
supported  on  it  by  a  horizontal  force  equal  to  half 
the  weight  of  the  body.  Find  the  coefficient  of  fric- 
tion between  the  body  and  the  plane. 

363.  Two  equal  weights  are  attached  to  a  string 
that  is  laid  over  the  top  of  two  inclined  planes  having 
the  same  altitude  and  placed  back  to  back,  the 
angles  of  inclination  being  30"  and  60°  respectively. 
Show  that  the  weights  will  be  on  the  point  of  moving 
if  the  coefficient  of  friction  between  each  plane  and 

I 
weight  be 


2+ V3 

364.  The  roughness  of  a  plane,  of 
inclination  30°  is  such  that  a  body  of 
weight  500  pounds  can  rest  on  it. 
What  is  the  least  force  required  to 
draw  the  body  up  the  plane  }  (a  in 
sketch  will  equal  the  angle  of  friction.) 


Fig.  46. 


365.  Find    the   least  force   that  will    drag  a    body 

weighing  100  pounds  along  a  rough  horizontal  plane, 

I 
the  coefficient  of  friction  being   — =•     Find  also  the 

resistant  reaction  of  the  plane. 

366.  A  weight  of  5  pounds  can  just  be  supported  on 
a  rough  inclined  plane  by  a  weight  of  2  pounds,  or  can 
just  support  a  weight  of  4  pounds  suspended  by  a 
string  passing    over  a  smooth  pulley  at   the  vertex. 


lOO 


ME  CHA  NICS-PR  OBLEMS. 


Find  the  coefficient  of  friction,  and  the  inclination  of 
the  plane. 

367.  A  heavy  cone  is  placed 
on  a  rough  inclined  plane,  the 
inclination  of  which  is  gradually 
increased.  Find  whether  the 
cone  will  begin  by  sliding  down 
the  plane  or  toppling  over. 

Fjg.  47.  Assume  first   that  the  equilibrium  is 

broken  by  the  body  toppling  over;  if 
this  does  not  require  too  great  a  value  of  the  coefficient  or  angle  of 
friction,  then  the  equilibrium  will  be  broken  in  that  way. 

Let  ABC  represent  the  vertical  section  of  the  cone,  CH  its  axis, 

wc 

G  its  center  of  gravity  ;  then  HG  = 

4 
Let  the  inclination  of  the  plane  be  such  that  the  vertical  through 
G  passes  through  A,  the  lowest  point  of  the  base.     If  the  cone  can 
rest  here  without  sliding,  then   the  slightest  increase  of    inclination 
will  cause  it  to  topple  over. 

Let  0  =  ACH  =  1^  vertical  angle  of  the  cone. 
The  forces  on  the  cone  are  the  weight  along  GA  ;  and  the  reac- 
tion of  the  plane. 

.*.  the  reaction  of  the  plane  is  along  AG. 
This  can  only  be  so  if  AG  makes  with  the  normal  to  the  plane 
an  angle  less  than  a,  the  angle  of  friction. 

.*.  AGH  must  be  <Ca 
or  tan  AGH  must  be  <^ 

AH 


But 


tan  AGH 
tan  ACH' 


HG 

ah' 

HC 


HC 
HG 


.-.  tan  AGH  =  4  tan  ACH  =  4  tan  0 

If  /A  >  4  tan  0,  the  cone  will  topple  over. 

If  /x  <^  4  tan  0,  the  cone  will  begin  to  slide,  and  its  motion  will 
start  as  soon  as  the  inclination  of  the  plane  is  a,  or  tan-i  /*. 


FOR  CES  —  FRIC  TION.  I O I 

368.  A  rectangular  block  ABCD  whose  height  is 
double  its  base,  stands  with  its  base  AD  on  a  rough 
floor,  coefficient  of  friction  \.  If  it  be  pulled  by  a 
horizontal  force  at  C  till  motion  ensues,  determine 
whether  it  will  slip  on  the  floor,  or  begin  to  turn  over 
round  D. 

369.  A  cubical  block  rests  on  a  rough  plank  with 
its  edges  parallel  to  the  edges  of  the  plank.  If, 
as  the  plank  is  gradually  raised,  the  block  turns  over 
on  it  before  slipping,  how  much  at  least  must  be  the 
coefficient  of  friction } 

370.  The  poles  supporting  a  lawn-tennis  net  are 
kept  in  a  vertical  position  by  guy-ropes,  one  to  each 
pole,  which  pass  round  pegs  2  feet  distant  from  the 
poles.  If  the  coefficient  of  friction  between  the  ropes 
and  pegs  be  |,  show  that  the  inclination  of  the 
latter  to  the  vertical  must  be  less  than  tan~^  -^^y  the 
height  of  the  poles  being  4  i^f^X. 

371.  The  table  of  a  small  planing-machine  which 
weighs  112  pounds,  makes  six  double  strokes  of  /\^ 
feet  each  per  minute.  The  coefficient  of  friction  be- 
tween the  sliding  surfaces  is  .07.  What  is  the  work 
in  foot-pounds  per  minute  performed  in  moving  the 
table } 

372.  A  rough  wedge  has  been  inserted  into  a 
block,  and  is  only  acted  on  by  the  reactions.  If  it 
is  on   the   point   of   slipping  out,  and   the  coefficient 

of  friction  is  — =,  what  is  the  angle  of  the  wedge  } 


1 02  MECHANICS-I'ROBLEMS. 

373.  A  cotter,  or  wedge,  having  a  taper  of  i  in  8, 
is  driven  into  a  cottered  joint  with  an  estimated 
pressure  of  600  pounds.  Taking  the  coefficient  of 
friction  between  the  two  surfaces  as  0.2,  find  the 
force  which  the  cotter,  or  wedge,  exerts  at  the  joint 
perpendicular  to  the  pressure  of  600  pounds ;  also 
find  the  pull  necessary  to  withdraw  the  cotter. 

374.  A  steel  wedge  1 2  inches  long,  tapered  from  2 
inches  thick  down  to  o,  is  used  to  wedge  up  a  pump 
plunger  weighing  3  000  pounds  by  means  of  a  maul 
weighing  5  pounds.  The  coefficient  of  friction  is  o.  i  5 
and  the  striking  velocity  of  the  maul  is  25  feet  per 
second.     How  far  will  each  blow  drive  the  wedge } 

375.  The  Idcomotive  of  the  Empire  State  Express 
has  four  drivers  and  a  total  weight  of  124000 
pounds ;  the  weight  on  the  drivers  is  84  000  pounds  ; 
the  coefficient  of  friction  between  wheels  and  rails  is 
0.18.  Find  the  greatest  pull  which  the  engine  can 
exert  in  pulling  itself  and  a  train.  What  is  the  total 
weight  of  itself  and  train  which  it  can  draw  up  a 
grade  of  i  in  100,  if  the  resistance  to  motion  is  12 
pounds  per  ton  on  the  level  1 

376.  A  wheel  of  weight  W  rests  between  two 
planes,  each  inclined  to  the  vertical  at  angle  a;  the  plane 
of  the  wheel  is  perpendicular  to  the  line  of  intersec- 
tion of  the  two  planes,  which  is  itself  horizontal.  If 
/x  be  the  coefficient  of  friction,  find  the  least  couple 
necessary  to  turn  the  wheel. 


FO  R  CES  —  FRIC  T/OJV, 


03 


M 

1     , 

!  !r'^ 

.Bi_ 

/    ]    / 

1    ■ 

/   /c 

1 

R   //  ^ 

1 

Fig.  48. 


377.  A  uniform  ladder  of  weight 
W  rests  on  rough  ground  and  against 
a  rough  wall,  the  coefficients  of 
friction  being  respectively  /x  and  fx!. 
What  is  the  least  inclination  it  can 
make  with  the  horizon  ? 


Let  AB  be  the  ladder,  AO  the  ground,  BO 
the  wall.    The  ladder  is  about  to  slip  down  ; 
therefore  the  limiting  friction  acts  at  A  toward  the  wall,  and  at  B 
upwards. 

Let  the  nornial  reaction  at  A  —  R, 
.-.  Friction  at  A  =  fxK. 
Similarly  we  have  R'  and  ^lR.'  at  B. 

The  three  forces  acting  are  the  weight  of  the  ladder  and  the  re- 
sultant reactions  at  A  and  B. 

These  three  forces  must  meet  in  a  point  M  vertically  above  C, 
W  the  weight  of  the  ladder  acting  at  the  middle  point  C. 
Resolve  the  forces  horizontally  and  vertically. 
W  =  R  +  /t'R' 
R'  =  /xR 
W  =  R  +  mm'R 
Take  moments  about  B, 

AO 


RX  AO 

=  W-^-+/.RxBO 

.-.  R 

W 

=  —  +  /iR  tan  ^ 

.-.  W 

=  2  R  [r  —/A  tan  e\ 

Placing 

this 

equal 

to  value 

of  W  found  above, 

R[i-^ 
.-.  I  — 

.-.  tan 

tan  ^]  =  R  +  /i/R 
1x1/  —  2  fx  tan  6 

378.    A  ladder  inclined  at  an  angle  of  60°  to  the 
horizon,  rests  with   one  end  on  rough  pavement,  and 


I04  MECHANICS-PROBLEMS. 

the  Other  end  against  a  smooth  vertical  wall ;  the 
ladder  begins  to  slide  down  when  a  weight  is  put  at 
its  middle  point.     Show  that  the  coefficient  of  friction 

379.  A  uniform  ladder  weighing  lOO  pounds  and 
52  feet  long  is  inclined  at  an  angle  of  45°  with  a 
rough  vertical  wall  and  a  rough  horizontal  plane.  If 
the  coefficient  of  friction  is  at  each  end  |,  how  far  up 
the  ladder  can  a  man  weighing  200  pounds  ascend 
before  the  ladder  begins  to  slip } 

380.  A  uniform  ladder  70  feet  long  is  equally  in- 
clined to  a  vertical  wall  and  the  horizontal  ground, 
both  rough ;  a  man  with  a  hod  —  weight  224  pounds 
—  ascends  the  ladder  which  weighs  448  pounds. 
How  far  up  the  ladder  can  the  man  ascend  before  it 
slips,  the  tangent  of  the  angle  of  resistance  for  the 
wall  being  1  and  for  the  ground  \  ? 

381.  A  uniform  beam  rests  with  one  end  on  a 
rough  horizontal  plane,  and  the  other  against  a  rough 
vertical  wall,  and  when  inclined  to  the  horizon  at  an 
angle  of  30°,  is  on  the  point  of  slipping  down  :  suppose 
the  surfaces  equally  rough,  find  /x. 

382.  The  mean  diameter  of  the  threads  of  a  | 
inch  bolt  is  .45  inches,  the  slope  of  the  thread  .07 
and  the  coefficient  of  friction  0.16.  Find  the  tension 
in  a  bolt  when  tightened  up  by  a  force  of  20  pounds 
on  the  end  of  a  wrench  1 2  inches  long. 


FORCES  —  FRIC  TION.  I O  5 

383.  Experiment  shows  that  a  weight  can  lift  only 
three-quarters  of  its  own  weight  by  means  of  a  rope 
over  a  single  pulley,  this  being  the  consequence  of 
the  stiffness  of  the  rope  and  the  friction  of  the  axis. 
Hence  show  that  the  mechanical  advantage  of  four 
such  pulleys  arranged  in  two  blocks  is  about  2.05. 

384  A  weight  of  5  tons  is  to  be  raised  from  the 
hold  of  a  steamer  by  means  of  a  rope  which  takes 
3I  turns  around  the  drum  of  a  steam-windlass.  If 
^  =  0.234,  what  force  must  a  man  exert  on  the  other 
end  of  the  rope  } 

logio  '1\  =  logio  T2  -f-  2 . 7  2  88  «/X. 

385.  A  man  by  taking  2^  turns  around  a  post  with 
a  rope,  and  holding  back  with  a  force  of  200  pounds, 
just  keeps  the  rope  from  surging.  Supposing  /x  = 
0.168,  find  the  tension  at  the  other  end  of  the  rope. 

386.  A  weight  of  2  000  pounds  is  to  be  lowered 
into  the  hold  of  a  ship  by  means  of  a  rope  which 
passes  over  and  around  a  spar  lashed  across  the  hatch- 
coamings  so  as  to  have  an  arc  of  contact  of  \\  cir- 
cumferences. If  /A  =  2V'  what  force  must  a  man 
exert  at  the  end  of  the  rope  to  control  the  weight  t 

387.  A  hawser  is  subjected  to  a  stress  of  10  000 
pounds.  How  many  turns  must  be  taken  around  the 
bitts,  in  order  that  a  man  who  cannot  pull  more  than 
250  pounds  may  keep  it  from  surging,  supposing 
ya  =  0.168.? 


io6 


ME  CHA  NICS-PROBLEMS. 


88.  A  rope  drive  carrying  20  ropes,  has  a  pulley 
16  feet  in  diameter,  and  transmits  600  horse-power 
when  running  at  90  revolutions  per  minute.  Taking 
/x  =  0.7  and  the  angle  of  contact  180°,  find  the  ten- 
sions on  the  tight  and  slack  sides  of  the  rope. 

Tj-T,  =  218.8 
T2  =  Ti-  218.8 
Ti=  (Ti-  2i8.8>'-^ 
Ti(^'^«—  i)  =  218.8^^^ 
.2i8.8^M« 


Ti  = 


218.8  X  2.72-7X1 

2.72-7X'^— I 


389.    A  single  fixed  pulley,  6  inches  in  diameter, 
turns  on  an  axle  2  inches  in  diameter  ;  coefficient  of 
friction    0.2.     A    weight  of   500  pounds 
is  lifted  by  means  of  this  pulley.      Find 
the  force  P  that  is  required. 

Taking  moments  about  C,  the  center 
—  P  X  3  +  500  X  3  +  /xR  X  I  =  o 
/u,  =  0,2 
S  =  R'  +  /^t'R' 
S  =  P  -f  W 
j^^P  +  W 


Fig.  49. 

Since 
and 


P  +  500 


1.02 


.'.  P  X  3  =  500  X  3  -f  0.2 


p-j-  500 

X —  X 


1.02 
P  =  570  i:cunds. 


FORCES  —  FRIC  TION.  I O/ 

390.  A  single  fixed  pulley,  2  feet  in  radius,  turns 
on  an  axle  i  inch  in  radius ;  the  weight  of  the  pulley 
is  80  pounds.  A  weight  of  500  pounds  is  lifted  by 
means  of  this  pulley  ;  What  force  P  is  required .? 
The  coefficient  of  friction  between  axle  and  bearing 
is  o.  I  ;  the  rope  is  supposed  to  be  flexible,  and  with- 
out weight,  and  P  to  act  vertically. 

391.  Let  P  and  W  be  inclined  to  each  other  at  an 
angle  of  90°  ;  radius  of  pulley  is  6  inches ;  radius  of 
axle  I  inch  ;  coefficient  of  friction,  0.2.  Determine 
the  relation  of  P  and  W  in  case  of  incipient  motion. 

392.  A  leather  belt  will  stand  a  pull  of  200  pounds. 
It  passes  around  one-half  the  circumference  of  a  pul- 
ley 4  feet  in  diameter  making  150  revolutions  per 
minute.  What  power  will  it  transmit  if  the  coefficient 
of  friction  between  the  belt  and  pulley  is  o.  i  } 

393.  Find  the  width  of  a  belt  necessary  to  transmit 
10  horse-power  to  a  pulley  12  inches  in  diameter,  so 
that  the  greatest  tension  may  not  exceed  40  pounds 
per  inch  of  width  when  the  pulley  makes  i  500  revo- 
lutions per  minute,  the  weight  of  the  belt  per  square 
foot  being  1.5  pounds,  and  the  coefficient  of  friction 
0.25. 

394.  A  belt  laps  150°  around  a  3-foot  pulley,  mak- 
ing 1 30  revolutions  per  minute  ;  the  coefficient  of 
friction  is  0.35.  What  is  the  maximum  pull  on  the 
belt  when  20  horse-power  is  being  transmitted  and 
the  belt  is  just  on  the  point  of  slipping  1 


I08  MECHANICS-PROBLEMS. 

395.  The  power  of  an  engine  is  tested  by  putting  a 
belt  over  the  fly-wheel,  which  is  5  feet  in  diameter, 
and  on  one  end  of  the  belt  hanging  a  weight  of  300 
pounds  and  to  the  other  attaching  a  spring  balance. 
The  fly-wheel  is  observed  to  make  150  revolutions 
per  minute  and  the  spring  balance  reads  180  pounds. 
What  is  the  brake  horse-power } 

396.  In  problem  395,  if  the  belt  laps  more  than 
180°  the  dynamometer  pulls  on  belt  being  300  pounds 
and  180  pounds,  coefficient  of  friction  0.159,  what 
part  of  the  circumference  is  encircled  1 

397.  A  12-inch  Pelton  water-motor  of  3  horse^ 
power  is  tested  by  a  friction  brake  encircling  three- 
fourths  of  the  4-inch  pulley  of  the  motor  and  having 
a  lever  arm  extending  22  inches  from  center  of  pul- 
ley to  scales.  The  scales  read  5  pounds  when  motor 
is  making  i  1 50  revolutions  per  minute.  What  horse- 
power IS  being  developed } 

398.  Find  the  horse-power  necessary  to  turn  a  shaft 
9  inches  in  diameter  making  75  revolutions  per  min- 
ute, if  the  total  load  on  it  is  12  tons  and  /x  ==  .015. 

_  s 

399.  A  shaft  makes  50  revolutions  per 
minute.  If  the  load  on  the  bearing  be  8 
tons  and  the  diameter  of  the  bearing  be  7 
inches,  at  what  rate  is  heat  being  generated, 
the  average  coefficient  of  friction  being 
0.05  } 


FOR  CES  —  FKIC  TION.  I OQ 

S  =  8  tons 
=  PH- W 
P  + W 


R  = 


8X2  ooo 


Vi.o  025 
16  000 


1. 001 
=  15980 
^R  =  799  pounds, 
7-inch  diameter  =  22-inch  circumference. 

2|  X  50  =  ^\%^  =  92  feet  per  minute 
Foot-pounds  generated  by  heat 

=  799  X  92 

=  73  500  foot-pounds  per  minute. 

400.  A  horizontal  axle  10  inches  in  diameter  has  a 
vertical  load  upon  it  of  20  tons,  and  a  horizontal  pull 
of  4  tons.  The  coefficient  of  friction  is  0.02.  Find 
the  heat  generated  per  minute,  and  the  horse-power 
wasted  in  friction,  when  making  50  revolutions  per 
minute. 

401.  A  horizontal  axle  10  inches  in  diameter  has 
upon  it  a  vertical  load  of  20  tons,  and  a  horizontal 
pull  of  4  tons.  The  coefficient  of  friction  is  0.02. 
Find  by  the  rough  method  given  below  the  heat  gen- 
erated per  minute,  and  the  horse-power  wasted  in 
friction  when  making  50  revolutions  per  minute,  tak- 
ing the  resistance  as  2  pounds  per  square  inch. 

(  "  A  rough  and  ready  estimate  of  the  work  ab- 
sorbed by  a  bearing  is  to  assume  that  the  frictional 


no  MECHANICS-PROBLEMS. 

resistance  of  the  surface  of  the  bearing  is  3  pounds 
per  square  inch  for  ordinary  kibrications,  2  pounds 
for  pad,  I  pound  for  bath,  the  surface  being  reck- 
oned on  the  nominal  area."  —  Goodman.) 

402.  The  shaft  of  a  i  000-kilowatt  dynamo  is  25 
inches  in  diameter,  makes  100  revokitions  per  min- 
ute, and  carries  a  total  load  of  45  000  pounds.  The 
coefficient  of  friction  being  0.05,  find  the  horse-power 
lost  in  heat  that  is  generated  by  friction. 

403.  Calculate  the  horse-power  absorbed  by  a  foot- 
step bearing  with  flat  end  8  inches  in  diameter  when 
supporting  a  load  of  4000  pounds,  and  making  ico 
revolutions  per  minute,  coefficient  of  friction  0.03. 

404.  Find  the  horse-power  absorbed  in  overcoming 
the  friction  of  a  foot-step  bearing  4  inches  in  diam- 
eter, the  total  load  being  i|  tons,  the  number  of  rev- 
olutions 100  per  minute,  and  the  average  coefficient 
of  friction  0.07. 

Moment  of  friction  =  f  /xWR     (See  text-books.) 
Work  per  minute      =  f  /uW  x  ^  X  2  tt  X  N 

(D  being  in  inches.) 

_/LtWDN 

TT 

_/u.WDN 

~     5-73 


MOTION.  Hi 


III.    MOTION 

405.  A  body  moving  with  a  velocity  of  5  feet  per 
second  is  acted  on  by  a  force  which  produces  a  con- 
stant acceleration  of  3  feet  per  second.  What  is  the 
velocity  at  the  end  of  20  seconds  ? 

Velocity    gained  =  acceleration   per  second  X  number  of 
seconds. 

v=fXt 
=  3  X  20 

=  60  feet  per  second 
Final  velocity  =  60-1-5 

=  65  feet  per  second. 

406.  The  initial  velocity  of  a  stone  is  1 2  feet  per 
second  ;  this  velocity  decreases  uniformly  at  the  rate 
of  2  feet  per  second.  How  far  will  the  stone  have 
traveled  in  5  seconds  .?  « 

407.  Two  trains  A  and  B  moving  towards  each 
other  on  parallel  rails  uniformly  at  the  rate  of  30 
miles  and  45  miles  an  hour,  respectively,  are  5  miles 
apart  at  a  given  instant.  How  far  apart  will  they  be 
at  the  end  of  6  minutes  from  that  instant,  and  at 
what  distances  are  they  from  the  first  position  of  A .? 

408.  The  velocity  of  a  train  is  known  to  have  been 
increasing  uniformly  ;  at  one  o'clock  its  velocity  was 
12  miles  per  hour,  at  10  minutes  past  one  its'  velocity 
was  36  miles  an  hour.  What  was  its  velocity  at  7^ 
minutes  past  one  "i 


MOTION.  113 

409.  A  train  moving  at  the  rate  of  30  miles  an 
hour  is  brought  to  rest  in  2  minutes ;  the  retardation 
is  uniform.      How  far  did  it  travel } 

410.  A  body  acted  on  by  a  constant  force  begins 
to  move  from  a  state  of  rest ;  it  is  observed  to  move 
through  55  feet  in  a  certain  2  seconds,  and  through 
"jj  feet  in  the  next  2  seconds.  What  distance  did  it 
describe  in  the  first  6  seconds  of  its  motion  } 

411.  A  stone  skimming  on  ice  passes  a  certain 
point  with  a  velocity  of  20  feet  per  second  and 
suffers  a  retardation  of  one  unit.  Find  the  space 
passed  over  in  the  next  10  seconds,  and  the  whole 
space  traversed  when  the  stone  had  come  to  rest. 

412.  An  ice  boat  weighing  i  000  pounds  is  driven 
for  30  seconds  from  rest  -by  a  wind  force  of  100 
pounds.  Find  the  velocity  acquired  and  the  distance 
passed  over. 

413.  Two  bodies  are  let  fall  from  the  same  point 
at  an  interval  of  2  seconds.  Find  the  distance  be- 
tween them  after  the  first  has  fallen  for  6  seconds. 

For  I  St  body,  s  =  ^  gi'^ 

=  I-  X  32  X  6^ 

=  576  feet. 
For  2d  body,  s  =  ^gt^ 

=  1x32x4' 
=  256  feet. 
.*.  distance  apart  =  576-256 
=  320  feet. 


I  1 4  MECHANICS-PROBLEMS. 

414.  A  Stone  is  projected  vertically  upwards  with 
a  velocity  of  80  feet  per  second  from  the  summit  of  a 
tower  96  feet  high.  In  what  time  will  it  reach  the 
ground,  and  with  what  velocity } 

415.  A  stone  is  dropped  into  a  well,  and  the  sound 
of  its  striking  is  heard  2^^  seconds  after  it  is  let  fall ; 
the  velocity  of  sound  in  air  is  i  200  feet  per  second. 
What  is  the  depth  of  the  well  1 

Let  s  =  depth  of  well. 

.'.   time  for  sound  to  come  up  =  — ^ —  seconds. 

I  200 

Time  for  stone  to  fall  is  found  from  formula 

2  S  S 

4       . 
Time  for  stcne  to  fall  -f  time  md  to  come  up  =  2^^. 


or 


I  200        4        12 
.-.  s  +  300  Vi"  =^3  100 
s  ±  150'^  +  300  V^  =  3  100  ±  150^ 
V^  =—  310  an  inadmissible  value, 

\G  =  +io 
J"  =  100  feet,  depth  of  well. 


416.  A  stone  is  let  fall  from  a  tower  of  height  a 
feet ;  another  is  projected  upwards  vertically  from  the 
foot  of  the  tower ;  the  two  start  at  the  same  moment. 


MO  TION.  I  I  5 

What  is  the  initial  velocity  of  the  second  if  they  meet 
halfway  up  the  tower  ? 

417.  A  stone  is  dropped  into  a  well,  and  the  sound 
of  the  splash  is  heard  j.y  seconds  afterwards.  Find 
the  depth  of  the  well,  supposing  the  velocity  of  sound 
to  be  I  1 20  feet  per  second. 

418.  A  bucket  is  dropped  into  a  well  and  in  4  sec- 
onds the  sound  of  its  striking  the  water  is  heard. 
How  deep  is  the  well  1 

419.  A  balloon  has  been  ascending  vertically  at  a 
uniform  rate  for  42  seconds,  and  a  stone  let  fall  from 
it  reaches  the  ground  in  7  seconds.  Find  the  velocity 
of  the  balloon  and  the  height  from  which  the  stone  is 
let  fall. 

420.  From  a  Walloon  vhich  is  ascending  with  a 
velocity  of  32  feet  p^'  :econd,  a  ball  is  let  fall  and 
reaches  the  ground'*  ''•'^^^7  seconds.  How  high  was 
the  balloon  when  th-e^  >ione  was  dropped  t 

421.  A  ball  is  let  f  ill  to  the  ground  from  a  certain 
height,  and  at  the  same  time  another  ball  is  thrown 
upwards  with  just  sufficient  velocity  to  carry  it  to  the 
point  from  which  the  first  one  fell.  Find  when  and 
where  they  will  meet. 

422.  A  cake  of  ice  slides  down  a  smooth  chute,  at 
an  angle  of  30°  to  the  horizon.  Through  how  many 
feet  vertically  will  it  fall  in  the  fourth  second  of  its 
motion } 


Il6  MECHANICS-PROBLEMS. 

Space  =  average  velocity  X  total  time 
Average  velocity  =  \  final  velocity   (for    constant   accel- 
eration) 
Final  velocity        =  gain  per  second  (the  acceleration)  x 
number  of  seconds 

Average  velocity  =  \ft 

.'.  space  =  \ft  X  t 


=  V/r' 

=  ^  gt^,  for  a  falling  body. 

J  =  J  X  32  X  4^  iri  four  seconds 

=  256  feet. 

s  =  \  X  32  X  3^  in  three  seconds 

=  144  feet. 

During  fourth  second, 

j-=  256-  144 

=  112  feet  on  incline. 

If  the  chute  has  inclination   of  30°  the   vertical   com- 

ponent of  distance  will  be 

s  —  112  X  sin  30° 

=  112  X  ^ 

=  56  feet. 

423.  A  cable  car  "  runs  wild  "  down  a  smooth  track 
of  inclination  20°  to  the  horizontal.  How  far  does  it 
go  during  the  first  8  seconds  after  starting  from 
rest } 

424.  A  velocity  of  6  V2  along  the  diagonal  of  a 
squarq  is  resolved  into  two  rectangular  components 
along  the  sides  of  the  square.  How  much  is  each 
component .? 


MOTION. 


117 


425.  A  body  is  sliding  with  velocity  //  down  an  in- 
clined plane  whose  inclination  to  the  horizon  is  30°. 
Find  the  horizontal  and  vertical  components  of  this 
velocity. 


426.  A  deer  is  running  at  the  rate  of  20  miles  an 
hour,  and  a  sportsman  fires  at  him  when  he  is  at  the 
nearest  point,  200  yards  distant.  How  many  feet  in 
advance  of  him  should  aim  be  taken  if  the  velocity  of 
the  bullet  be  i  000  feet  per  second } 

427.  A  boat  is  rowed  at  the  rate 
of  5  miles  an  hour  on  a  river  that 
runs  4  miles  an  hour.  In  what  di- 
rection must  the  boat  be  pointed 
to  cross  the  river  perpendicularly } 
With  what  velocity  does  it  move  t 

X 
Let    OX  be  4  units  in  length  to  represent  Fig.  51. 

the  velocity  of  the  stream. 

Draw  OM  perpendicular  to  OX.     The  resultant  velocity  is  to  be 

in  the  direcdon  OM. 

With  center  X  and  radius  of  5  units  describe  an  arc  cutting  OM 

in  P. 

Join  XP,  and  complete  the  parallelogram  of  velocities  OXPQ. 

OQ  is  the  required  direction. 

The  angle  QOP  =  sin  - 1  i. 

Therefore  the  boat   must   not  be  rowed  straight  across,  but  up 

stream  at  an  angle  of  53°  10'. 

To  find  the  resultant  velocity. 

OP2  =  OQ2-QP2 

=  52-42 

=  25-16 

=  9 
.•.OP  =  3 


1  I  8  MECHANICS-PROBLEMS. 

428.  A  river  flows  at  the  rate  of  2  miles  per  hour. 
A  boat  is  rowed  in  such  a  way  that  in  still  water  its 
velocity  would  be  5  feet  per  second  in  a  straight  line. 
The  river  is  3  000  feet  wide ;  the  boat,  starting  from 
one  shore,  is  headed  60°  up-stream.  Where  will  it 
strike  the  opposite  shore  t 

429.  A  bullet  moving  upwards  with  velocity  of 
I  000  feet  per  second,  hits  a  balloon  rising  with  velo- 
city 100  feet  per  second.      Find  the  relative  velocity. 

430.  A  train  at  45  miles  an  hour,  passes  a  carriage 
moving  10  yards  a  second  in  the  same  direction  along 
a  parallel  road.      Find  the  relative  velocity. 

431.  To  a  passenger  in  a  train,  raindrops  seem  to 
be  falling  at  an  angle  of  30°  to  the  vertical  ;  they  are 
really  falling  vertically,  with  velocity  80  feet  per 
second.     What  is  the  speed  of  the  train  } 

432.  Two  roads  cross  at  right  angles  ;  along  one 
a  man  walks  northward  at  4  miles  per  hour,  along  the 
other  a  carriage  goes  at  8  miles  per  hour.  What  is 
the  velocity  of  the  man  relative  to  the  carriage } 

433.  A  steamer  is  proceeding  E.  with  a  velocity  of 
6  miles  per  hour ;  the  wind  appears  to  blow  from  the 
N.;  the  steamer  increases  its  velocity  to  12  miles  per 
hour,  and  the  wind  now  appears  to  blow  from  the  N.  E. 
What  is  the  true  direction  of  the  wind  and  its 
velocity  } 

434.  A  ship  is  sailing  north-east  with  a  velocity  of 
10  miles  per  hour,  and  to  a  passenger  on  board  the 


MOTION.  119 

wind  appears  to  blow  from  the  north  with  a  velocity 
of  10  V2  miles  per  hour.  Find  the  true  velocity  of 
the  wind. 

435.  Two  trains,  whose  lengths  are  respectively 
130  feet  and  1 10  feet,  moving  in  opposite  directions 
on  parallel  rails,  are  observed  to  be  4  seconds  in  com- 
pletely passing  each  other,  the  velocity  of  the  longest 
train  being  double  that  of  the  other.  Find  at  what 
rate  per  hour  each  train  is  moving. 

436.  A  fly-wheel  revolves  1 2  times  a  second.  What 
is  the  angular  velocity  about  the  center  of  a  point  on 
its  rim  1 

437.  A  train  weighing  60  tons  has  a  velocity  of 
40  miles  an  hour  when  the  steam  is  shut  off.  If 
the  resistance  to  motion  is  10  pounds  per  ton,  and 
no  brakes  are  applied,  how  far  will  it  have  traveled 
when  the  velocity  has  reduced  to  10  miles    per  hour  .<* 

438.  A  freight  train  of  100  tons  weight  is  moving 
at  the  rate  of  30  miles  per  hour  when  the  steam  is 
shut  off  and  the  biakes  applied  to  the  locomotive. 
Supposing  the  only  friction  is  that  at  the  locomotive, 
the  weight  of  which  is  20  tons,  what  is  the  coefficient 
of  friction  if  the  train  stops  after  moving  2  miles  } 

439.  A  steamer  approaching  a  dock  with  engines 
reversed  so  as  to  produce  a  uniform  retardation  is 
observed  to  make  500  feet  during  the  first  30  seconds 
of  the  retarded  motion  and  200  feet  during  the  next 


1 2  O  ME  CIIA  NICS-PR  OBL  EMS. 

30  seconds.      In   how  many  more    seconds    will  th& 
headway  be  completely  stopped  ? 

440.  A  ball  is  thrown  along  a  rough  floor,  coeffi- 
cient of  friction  |.  What  will  be  its  velocity  after  3 
seconds,  if  the  original  velocity  is  50  feet  per  second  ? 

441.  A  body  is  projected  up  an  inclined  plane,  of 
angle  30°,  with  a  velocity  of  80  feet  per  second. 
Find  — 

(i)  How  long  it  will  be  before  coming  to  rest? 

(2)  How  far  it  will  go  up  the  plane  ? 

(3)  How  long  it  will  be  in  returning  to  its  starting- 

point  ? 

(4)  With  what  velocity  will  it  return  to  its  starting- 

point  ? 

442.  A  cannon  when  fired  recoils  with  a  velocity 
of  10  feet  per  second  and  runs  up  a  platform  having 
an  incline  of  i  in  4.  Find  the  horizontal  distance  it 
goes  before  coming  to  rest. 

443.  A  locomotive  weighing  100  tons  is  observed 
to  be  increasing  its  speed  at  the  rate  of  100  feet  a 
minute.     What  is  the  effective  force  acting  on  it  t 

444.  What  force  must  be  exerted  by  an  engine  to 
move  a  train  of  mass  100  tons  with  10  units  of  accel- 
eration, if  frictional  resistances  are  5  pounds  per 
ton  I 

445.  A  train  of  100  tons,  excluding  the  engine, 
runs  up  a  I  %  grade  with  an  acceleration  of  i  foot  per 


MOTION.  121 

second.     If  the  friction  is   lo  pounds  per  tons,  find 
the  pull  on  the  drawbar  between  engine  and  train. 

446.  A  force  of  5  pounds  is  made  to  move  a  body 
weighing  50  pounds.  What  is  the  acceleration  pro- 
duced } 


<b 


Power  producing  motion  :  whole  mass  moved  =/" 

5  •  50  =7:32 

50/=  5  X  32 

y=  3I- feet  per  second. 

447.  A  body  moving  along  a  straight  line  is  known 
to  be  acted  on  by  a  constant  force  ;  at  a  certain  in- 
stant it  is  moving  at  the  rate  of  12  feet  a  second, 
and  in  the  next  10  seconds  it  describes  a  distance  of 
470  feet.  What  velocity  does  it  gain  in  each  second 
of  its  motion  } 

s  =  Vt  +  iy/2 
470  =  12  X  10  +  i/io2 
350  =  i/X  100  • 

=  7  feet  per  second. 

448.  A  body  whose  mass  is  108  pounds  is  placed 
on  a  smooth  horizontal  plane,  and  under  the  action  of 
a  certain  force  describes  from  rest  a  distance  i  li  feet 
in  5  seconds.     What  is  the  force  acting .? 

449.  Masses  of  5  pounds  and  1 1  are  connected 
by  a  weightless  thread;  the  ii-pound  weight  is 
placed  on  a  smooth  horizontal  table,  while  the  other 
hangs  over  the  edge.      If  both  are  then  allowed  to 


I  2  2  MECHA NICS-PKOBLEMS. 

move  under  the  action  of  gravity,  what  is  the  tension 
of  the  thread  ? 

450.  A  lo-pound  weight  hangs  over  the  edge  of  a 
table  and  pulls  a  45-pound  box  along ;  the  coefficient 
of  friction  between  the  table  and  the  box  is  0.05. 
Find  the  acceleration  and  the  tension  in  the  string. 

451.  The  table  of  a  box-machine  weighs  50  pounds 
and  is  pulled  back  to  its  starting  position,  a  distance  of 
6  feet,  by  a  falling  weight  of  20  pounds.  What  time, 
neglecting  friction,  will  thus  be  used  in  return 
motion  } 

452.  A  weight  of  10  pounds  rests  6  feet  from  the 
edge  of  a  smooth  horizontal  table  that  is  3  feet  high. 
A  string  7  feet  long  passes  over  a  smooth  pulley  at 
the  edge  of  the  table  and  connects  with  a  lo-pound 
weight.  If  this  second  weight  is  allowed  to  fall,  in 
what  time  will  it  cause  the  first  weight  to  reach  the 
edge  of  the  table  } 

453.  A  balloon  is  moving  upward  with  a  speed 
which  is  increasing  at  the  rate  of  4  feet  per  second 
in  each  second.  Find  how  much  the  weight  of  a 
body  of  10  pounds  as  tested  by  a  spring  balance  on  it, 
would  differ  from  its  weight  under  ordinary  circum- 
stances. 

454.  A  man  who  is  just  strong  enough  to  lift  1 50 
pounds  can  lift  a  barrel  of  flour  of  200  pounds  weight 
when  going  down  on  an  elevator.  How  fast  is  the 
velocity  of  elevator  increasing  per  second } 


MOTIOA\  123 

455.  An  elevator,  starting  from  rest,  has  a  down- 
ward acceleration  of  |  g  for  i  second,  then  moves 
uniformly  for  2  seconds,  then  has  an  upward  acceler- 
ation of  i  ^  until  it  comes  to  rest,  [a)  How  far 
does  it  descend  .?  {b)  A  person  whose  weight  is  140 
pounds  experiences  what  pressure  from  the  elevator 
during  each  of  the  three  periods  of  its  motion .? 

456.  If  a  train  ascends  by  its  own  momentum  a 
grade  of  i  in  40  for  a  distance  of  i  mile,  the  resist- 
ance from  friction,  etc.,  being  10  pounds  per  ton,  find 
its  initial  velocity. 

Work  =  2  000  X  132  =264  000  foot-pounds 
IP  X  5280  =  /Y^^r  fgg  foot-pounds 
1-  m7^  =  316  800 
i  X  ^%%^  Xv^  =  316800 
7^^  =  10  100 
V  =  100  -h,  feet  per  second, 

or  the  316  800  foot-pounds  could  be  accomplished  by 
the  I  ton  starting  with  a  velocity  acquired  by  falling 
from  a  height,  h. 

2  000  X  /^  =  316  800 

h  =  158.4. 
If  a  body  fell  from  this  height  its  velocity  would 


be 


=  8VT^ 

=  100  -f-,  feet  per  second  as  above. 


457.  A  particle  whose  mass  is  10  pounds  moves 
along  a  horizontal  plane  against  a  friction  of  one-fifth 
of  its  weight  for  a  distance  of  20  feet  before  coming 


1 24  ME  CHA  NICS-PR  OBLEMS. 

torest.     What    must    have  been    its  velocity  at    the 
beginning  of  the  20  feet  ? 

458.  Some  railroad  cars  start  from  rest  down  an 
incline  a  mile  long  with  a  gradient  of  i  in  100. 
Find  how  many  yards  they  will  travel  on  the  level, 
after  leaving  the  incline,  before  they  come  to  rest, 
the  frictional  resistance  to  their  motion  being  10 
pounds  per  ton.  Find  also,  in  miles  per  hour,  the 
greatest  velocity  that  they  will  acquire. 

459.  Two  weights  of  120  and  100  pounds  are  sus- 
pended by  a  fine  thread  passing  over  a  fixed  pulley 
without  friction.  What  space  will  either  of  them 
pass  over  in  the  third  second  of  their  motion  from 
rest  ? 

460.  A  cord  passing  over  a  smooth  pulley  carries 
10  pounds  at  one  end  and  54  pounds  at  the  other. 
What  will  be  the  velocity  of  the  weight  5  seconds 
from  rest,  and  what  will  be  the  tension  in  the  cord  } 

461.  An  engine  draws  a  three-ton  cage  up  a  coal- 
pit shaft  at  a  speed  uniformly  increasing  at  the  rate 
of  5  feet  per  second.  What  is  the  tension  m  the 
rope } 

=  937-5  pounds 
Pull  =  937.5  +  6  000 
=  6  937-5  pounds. 


MOTION.  125 

462.  Two  strings  pass  over  a  smooth  pulley  ;  on 
one  side  both  strings  are  attached  to  a  weight  of  5 
pounds,  on  the  other  side  one  string  is  attached  to  a 
weight  of  3  pounds,  the  other  to  one  of  4  pounds. 
Find  the  tensions  during  motion. 

463.  A  body  is  projected  with  a  velocity  of  50  feet 
per  second  in  a  direction  inclined  40°  upward  from 
the  horizontal.  Determine  the  magnitude  and  direc- 
tion of  the  velocity  at  the  end  of  2  seconds  {g  being 
taken  equal  to  32.15). 

Time  to  reach  highest  point, 

//  sin  a 

~       g 

^  50  X  .643  ,, 

32.15 

=  I  second. 

Therefore  in  another  second  the  body  will  fall  and 
be  in  position  similar  to  its  initial. 

464.  A  body  is  projected  with 
a  velocity  of  20  feet  per  second 
down  a  plane  whose  inclination  is 
25"";    the   coefficient    of   friction  Fig.  52. 
being    0.4.      Determine    the    space    traversed    in 
seconds. 

Power  producing  motion  :  total  mass  moved  =/:  ^'" 
•423-.3625  :  i  =/:  32 

y  =  1.93  feet  per  second 

=  20  X  2  +  ^  X   1.93  X  22 
=  43.9  feet. 


126  MECHANICS-PROBLEMS 

465.  A  body  slides  down  a  rough  inclined  plane  lOO 
feet  long,  the  sine  of  whose  angle  of  inclination  is  0.6 
and  coefficient  of  friction  is  |.  Find  the  velocity  at 
the  bottom.  If  projected  up  the  plane  with  a  velo- 
city which  just  carries  it  to  the  top,  find  that  velocity 
and  the  height  it  would  reach  if  thrown  vertically  up- 
wards with  the  same  velocity. 

466.  A  bullet  is  fired  with  a  velocity  of  i  000 
feet  per  second.  What  must  be  the  angle  of  inclina- 
tion, in  order  that  it  may  strike  a  point  in  the  same 
horizontal  plane,  at  a  distance  of  15  625  feet } 

467.  From  the  top  of  a  tower  a  stone  is  thrown  up 
at  an  angle  of  30°,  with  a  velocity  of  288  feet  per 
second  ;  the  height  of  the  tower  is  160  feet.  Find 
the  time  required  for  the  stone  to  reach  the  ground, 
and  how  far  it  will  have  gone  from  the  foot  of  the 
tower. 

468.  From  a  train  moving  at  60  miles  per  hour  a 
stone  is  dropped  ;  the  stone  starts  at  a  height  of  8 
feet  above  the  ground.  What  is  the  horizontal  dis- 
tance through  which  the  stone  has  gone  while  falling  } 

469.  From  a  quarry  blast  a  stone  has  a  velocity  of 
200  feet  per  second,  in  a  direction  inclined  at  an 
angle  of  60°  to  the  horizontal  plane.  To  what  height 
will  it  rise,  and  how  far  away  will  it  strike  the  ground  t 

470.  A  bullet  is  fired  with  a  velocity  of  which  the 
horizontal  and  vertical  components  are  80  and  120 
feet  per  second  respectively.  Find  the  range  and 
greatest  height. 


Time  of  flight  = 


MOTION.  xzy 

2  fx  sin  a 


2    X  I20 


32 

=  3_Q.  =  j^  seconds. 

Range  =  80  x  7^ 

=  600  feet 

1202 
Greatest  height  =  -^ — 

=  225  feet. 

471.  A  ball  is  discharged  with  the  initial  velocity 
of  I  100  feet.  How  many  miles  is  the  greatest  pos- 
sible range  ? 

472.  A  cannon  ball  is  fired  horizontally  from  a  hill 
that  is  on  the  coast  and  900  feet  high :  neglecting 
the  resistance  of  the  atmosphere,  find  the  time  which 
elapses  before  it  strikes  the  sea. 

473.  A  projectile  is  fired  horizontally  from  the  top 
of  a  hill  300  feet  high  to  a  ship  at  sea.  Its  initial 
velocity  is  2  000  feet  per  second  and  its  weight  500 
pounds.  What  will  be  its  range,  and  what  will  be 
the  energy  of  the  blow  which  it  strikes }  Neglect 
resistance. 

474.  The  top  of  a  fortification  wall  is  50  feet  above 
the  level  of  a  city.  From  a  man-of-war  in  the  bay 
250  feet  below  the  top  of  the  wall  and  distant  hori- 
zontally 3  000  feet,  a  projectile  is  fired  with  velocity 
of  I  000  feet  per  second.  The  projectile  just  clears 
the  wall.     Where  will  it  land  inside  the  city  ? 


12} 


MECHANICS-PROBLEMS. 
D 


Find    the  distance    AC   and  the  angle    B.     Then 
consider  flight  from  A  to  C. 


Horizontal  range  = 


=  ■\/3  ooo^  -h  300'^  = 


?/^sm  2C 


I  ooc^  X  sin  2  c 


32 


from  which  C  may  be  found  and 
Compute  //,  greatest  height  ;  subtract  250,  to  find  d. 

Find  / ;  then  find  the  horizontal,  /.  This  added  to 
one-half  the  range  will  give  the  position  where  pro- 
jectile will  land. 

(This  problem  may  well  be  done  by  methods  of 
calculus.) 

475.  A  rifle  projects  its  shot  horizontally  with  a 
velocity  of  i  000  feet  per  second  ;  the  shot  strikes  the 
ground  at  a  distance  of  i  000  yards.  What  is  the 
height  of  the  rifle  above  the  ground  } 


MOTION.  129 

476.  What  is  the  pressure  exerted  horizontally  on 
the  rails  by  an  engine  of  20  tons  weight  going  round 
a  curve  of  600  yards  radius  at  30  miles  an  hour  ? 

Velocity  =  44  feet  per  second. 


pressure  =  

20X2  000  X  44c 


pounds  weight 


I  800  X  32 
=  I  344  pounds. 

477.  A  train  of  60  tons  weight  is  rounding  a  curve 
of  radius  one  mile,  with  a  velocity  of  20  miles  an 
hour.      What  is  the  horizontal  pressure  on  the  rails  ? 

478.  An  engine  of  mass  24  tons  is  moving  round  a 
curve  cf  400  yards  radius,  and  the  horizontal  pres- 
sure exerted  on  the  rails  is  4.84  tons  weight.  What 
is  the  velocity  of  the  engine  ? 

479.  The  mass  of  the  bob  of  a  conical  pendulum  is 
2  pounds,  the  length  of  the  string  is  3  feet,  the  angle 
of  inclination  to  vertical  is  45°.     What  is  the  tension  t 

480.  The  mass  of  the  bob  is  20  pounds,  the  length 
of  the  string  is  2  feet,  the  tension  of  the  string  is 
5007r2  pounds  weight.  How  many  revolutions  per 
second  is  the  pendulum  making  t 

481.  If  a  conical  pendulum  be  10  feet  long,  the 
half  angle  of  the  cone  30°,  and  the  mass  of  the  bob 
12  pounds,  find  the  tension  of  the  thread  and  the 
time  of  one  revolution. 

482.  A  weight  of  10  pounds  is  fastened  by  a  string 
which  passes  through  a  hole  in  a  smooth  horizontal 


1 3  O  ME  CHA  NICS-PK  OBL  EMS. 

table  to  a  weight  of  i  pound,  which  hangs  vertically  ; 
the  first  weight  is  revolving  on  the  table  about  the 
hole  as  a  center.  How  many  revolutions  are  there 
per  minute  if  the  horizontal  portion  of  the  string  is 
1 5  inches  long  ? 

483.  A  ball  is  hung  by  a  string  in  a  passenger  car 
which  is  rounding  a  curve  of  i  ooo  feet  radius,  with 
a  velocity  of  30  miles  an  hour.  Find  the  inclination 
of  the  string  to  the  vertical. 

484.  A  pendulum  of  length  156.556  inches  oscil- 
lates in  two  seconds  at  London.  What  is  the  value 
of.-? 

485.  Given  /the  length  of  a  simple  pendulum,  tti/  |; 

the  time  of  an  oscillation  :  show"  how  to  find  approx- 
imately the  height  of  a  mountain  when  a  seconds 
pendulum,  by  being  taken  from  sea  level  to  its  sum- 
mit, loses  n  beats  in  24  hours.  If  ;/=  15,  what  is  the 
height  of  the  mountain,  the  radius  of  the  earth  being 
4  000  miles  t 

486.  Water  is  flowing  in  a  service  pipe  at  tne  rate 
of  24  feet  per  second ;  the  pipe  is  50  feet  long.  If 
the  water  be  uniformly  shut  off  by  a  stop  valve  in 
one-tenth  of  a  second,  show  that  the  water  pressure  in 
the  pipe  near  the  valve  is  increased  by  162.5  pounds 
per  square  inch. 

487.  If  in  the  above  pipe  the  pressure  of  the 
**  water  hammer  "  had  been  400    pounds   per  square 


MO  TIOjV.  I  3  I 

inch,  with  what  velocity  would  water  have  been  flow- 
ing at  the  beginning  ? 

488.  A  cricket  ball  of  mass  6  ounces  is  struck  so 
that  its  velocity  is  changed  from  lO  feet  per  second 
in  one  direction  to  20  feet  per  second  in  the  opposite. 
What  was  the  impulse  ? 

489.  A  hammer  of  10  tons  weight  falling  from  a 
height  of  4  feet  drives  a  wooden  pile  and  comes  to 
rest  in  3L  second.  How  far  does  it  drive  the  pile  ? 
And,  assuming  the  force  is  uniform,  find  it  and  the 
impulse. 

490.  An  8-inch  projectile  weight  250  pounds, 
strikes  a  sand  butt  with  velocity  of  2  000  feet  per 
second  and  is  stopped  in  25  feet.  If  the  resistance  is 
uniform,  what  is  its  value  in  pounds,  and  how  long  did 
it  take  to  stop  the  projectile  t 

491.  A  shot  of  mass  20  pounds  is  fired  from  a  gun 
of  mass  2000  pounds,  and  length  10  feet;  the  gun 
rests  at  the  foot  of  an  inclined  plane,  rising  i  in  15. 
If  the  muzzle  velocity  of  the  shot  be  i  200  feet  per 
second,  how  far  up  the  plane  will  the  gun  recoil  } 

492.  An  8-hundred  weight  shot  leaves  a  40-ton  gun 
with  velocity  of  2  000  feet  per  second:  the  length  of 
the  gun  is  20  feet.  What  is  the  average  force  of 
the  powder.? 

493.  A  man  weighing  160  pounds  jumps  with  a 
velocity  of   i6|  feet  per  second  into  a  boat  weighing 


I  3  2  ME  CHA  NICS-PK  OBL  EMS. 

lOO  pounds.     With  what  velocity  will  the  boat  move 
away  ? 

494.  An  8oo-pound  shot  is  fired  from  an  8i-ton 
gun,  with  a  muzzle  velocity  of  i  400  per  second  :  a 
steady  resistance  of  9  tons  begins  to  act  immediately 
after  the  explosion.     How  far  will  the  gun  move .'' 

495.  A  one-ounce  bullet  fired  out  of  a  20-pound 
rifle  pressed  against  a  mass  of  180  pounds,  kicks  the 
latter  back  with  an  initial  velocity  of  6  inches  per 
second.     Find  the  initial  velocity  of  the  bullet. 

Momentum  forward  =  momentum  backward 
MV  =  mv 
^1^  X  V  =  (180  +  20)  X  i 

V  =  I  600  feet  per  second. 

496.  A  ball  of  mass  4  pounds  and  velocity  4  feet 
per  second  meets  directly  a  ball  of  mass  5  pounds 
with  opposite  velocity  of  2  feet  per  second;  e  =  \. 
Find  the  velocities  after  impact. 

497.  A  freight  train,  weighing  200  tons,  and  travel- 
ing 20  miles  per  hour,  runs  into  a  passenger  train  of 
50  tons  standing  on  the  same  track.  Find  the  velo- 
city at  which  the  broken  cars  of  the  passenger  train 
will  be  forced  along  the  track,  supposing  e  =  \. 

498.  A  shell  bursts  into  two  fragments,  whose 
weights  are  12  and  20  pounds.  The  former  travels 
onward  with  a  velocity  of  700  feet  per  second,  and 
the  latter  with  a  velocity  of   380  feet  per  second. 


MOTION.  133 

What  was  the  momentum  of  the  shell  when  the  ex- 
plosion occurred  ? 

499.  There  are  two  bodies  whose  masses  are  15 
pounds  and  20  pounds  respectively  ;  the  former,  mov- 
ing at  the  rate  of  12  feet  a  second  overtakes  and 
impinges  directly  on  the  latter  moving  at  the  rate  of 
6  feet  a  second.  Find  their  common  velocity  at  the 
end  of  compression  and  their  joint  energies  just  before 
impact  and  at  the  end  of  compression.' 

500.  A  body  A  weighing  10  pounds,  and  moving 
at  the  rate  of  1 5  feet  a  second,  strikes  another  body 
B  weighing  20  pounds,  and  moving  at  the  rate  of  10 
feet  a  second,  in  the  direction  at  right  angles  to  that 
of  A's  motion.  The  bodies  are  to  be  treated  as  points, 
and  the  impact  is  supposed  to  take  place  in  the  direc- 
tion of  A's  motion.  Find  the  velocities  and  directions 
of  the  motions  of  the  bodies  after  impact,  the  restitu- 
tion being  perfect  (coefficient  of  elasticity  =  i ). 


A   FEW    IMPORTANT    UNIT    VALUES    TO 

BE  USED  IN  -SOLVING  THESE 

PROBLEMS 


I  hundred 

weight 

=  100  pounds 

I  ton 

=  2  000  pounds 

I  fathom 

=  6  feet 

I  knot 

=  6  080  feet 

I  cubic  foot  of  water 

=  62^  pounds 

=  yh  gallons 

I  gallon  of 

water 

=  84  pounds 

I  British  thermal  unit 

=  778  foot-pounds  of  energy 

g-,  accelleration  of  gravity  =  32  feet  per  second,  unless  other- 

wise  specified 

I  horse-power 

=  746  watts 

I  kilowatt 

=  1.34  horse-power 

Watts  =  volts  X  amperes 

TRIGONOMETRIC    FUNCTIONS 

0° 

30° 

45°             60°             90°             120° 

I 

i      ^3       ,       \r3 

Sine            0 

2 

V2 

.500 

.707           .866                             .866 

Vi 

I               I                                     I 

Cosine        i 

2 

V2             2                 °                  2 

.866 

.707           .500                        -  .500 

I 

V3 

I                ^^3         Infinite     —  Vs 

Tangent    0 

•577 

1.732                        -1.732 

Perp 

Base                                  Perp 

S^"    =  Hypot 

Cos - 

~  Hypot                   ^^'^  ~  Base 

Sin 

I                                          I 

T^"=Cos 

Cot  ^ 

~  Tan                       "*^"  ~  Cos 

Cosec  =  g^ 

Vers  = 

=  I  —  cos                 a:b  —  Sin  A  :  Sin  B 

Sin(^  +  ^)  =  Sin^Cos^+Cos^  Sin  .5     c  =  \ld^-^  U^- zab^ZosC 

134 


ANSWERS    TO    PROBLEMS 


3. 

3  360  foot-pounds. 

70. 

1 2  miles  an  hour. 

7. 

125  pounds. 

72. 

179.2      horse-power  ;     224 

10. 

2  704  000  foot-pounds. 

horse-power. 

11. 

97  500  foot-pounds. 

73. 

5  728    feet  ;    2  545.8    feet ; 

12. 

198  000  oco  foot-pounds. 

31.8  miles  per  hour;  27.3 

17. 

104  8  foot-pounds. 

miles  per  hour. 

22. 

About  13I  tons  weight. 

74. 

I  000  horse-power. 

27. 

20  ODO  foot-pounds. 

75. 

87  horse-power. 

28. 

266I  pounds. 

80. 

208  pounds. 

30. 

no  foot  pounds. 

81. 

140  horsepower. 

32. 

i|^  inches. 

82. 

107  horse-power. 

33. 

603!  :  I. 

87. 

5-7- 

31. 

.54  pounds. 

89. 

2  540  looms. 

35. 

30177- 

91. 

0.061  nearly. 

36. 

28 L  pounds. 

98. 

650  horse-power. 

37. 

120  pounds. 

100. 

5.6  horse-power. 

38. 

1 1 2  pounds. 

111. 

(i)  .9  cubic  feet;  {^)  1.69 

39. 

71 1  pounds;  301^. 

tons  ;     (3)     262.4     foot- 

40. 

522^  pounds. 

pounds;    (4)  39.5  horse- 

42. 

6  000  foot-pounds  ;  3:2. 

power. 

43. 

161    pounds;     11  800    foot- 

117. 

62.5  cubic  feet. 

pounds. 

125. 

0.14  horse-power;  97  cubic 

44. 

14  4  man -power. 

inches. 

45. 

324  500  foot-pounds;  41  per 

127. 

37!  pounds. 

cent;   226  units. 

88'a  R 

47. 

4  400  foot-pounds  ;  ^^  horse- 

129. 

— - —  strokes  per  minute. 

48. 
50. 
51. 

power, 
li  horse  power. 
36/Y  horse-power. 
21  horse-power. 
403j\  miles  per  hour. 
68    horse  power ;    303    am- 

130. 
131. 
135. 

1 1.2  pounds  per  ton. 

367  pounds;   19.5  per  cent. 

576  horse-power. 

57. 
60. 

140. 
141. 
142. 

871  horsepower.     • 
15  000  foot  pounds. 
3  inches, 
0.17  horse  power. 

63. 

peres. 
About  80  pounds  per  inch  of 

143. 

width. 

145. 

13  400  foot-pounds;  3  045.6 

66. 

4  inches. 

momentum  ;    i  522.8 

69. 

2T\   miles    per   hour ;   2y^o 

pounds ;  88  feet. 

miles;  6  minutes,  25  sec- 

150. 

yi^  second. 

onds. 

151. 

15  528  feet. 

135 


136 


ANSWERS. 


153.  lo  feet    per  second ;    shot 

does  more  work  as  1 62  :  i . 

154.  6  feet  3  inches. 

155.  265  pounds. 

158.  48  foot-pounds;  8  V^  feet 
per  second. 

160.  155  foot-pounds;  energy  at 
lowest  point  45  foot- 
pounds, 

163.  1 7 1  g^  revolutions ;  236  revo- 

lutions. 

164.  55    700   foot-pounds;    44.2 

turns. 

169.  20  foot-pounds. 

170.  20  feet. 

171.  (both)  500  foot-pounds. 

(A)  \  (500)  foot-pounds. 
(D)  I  (500)  foot-pounds. 

172.  cos^=-^. 

2  W 

173.  90  degrees. 

174.  100  pounds. 

176.  ID  pounds. 

177.  A  force  of  92.1  pounds  at 

angle  32°4o'  with  force 
of  44  pounds. 

178.  25  V3;  25. 
181.    29  pounds. 

183.  150  pounds,  90, 

184.  133^  pounds,  i66|. 

185.  ID  V^  pounds;  10. 

186.  6    tons   on   the    tie ;    6.32 

tons  on  the  rafters. 

187.  6f  tons. 

190.  416  pounds,  208. 

191.  50  pounds. 

193.  580  pounds. 

194.  6  030  pounds. 

195.  36  pounds,  164. 

196.  1 1 7. 1  pounds,  82.8. 

198.  \[^  i^oviwdi^,  2  \/5^. 

199.  2%  pounds,  9|. 

200.  cos3^= -;  (^  being  distance 

a 
from   C  to  AB,  2  a  the 
length  of  rod. 

203.  Perpendicular  to  plane. 

204.  500  pounds. 


205. 
206. 
207. 
208. 
212. 
213. 
215. 
216. 
218. 


223. 
225. 

226. 
229. 


I  020  pounds,  I  000. 
475  pounds. 
8.657  pounds. 
85  pounds,  iii. 
100  pounds,  100  ^/3. 
86.6  pounds,  100. 
20  pounds,  15. 
36.4  pounds,  27.3. 
77  pounds. 

1)  being  the  area  of  the  tri- 
angle, P  =  -— ^  (^2  4-  ^2 

1.16  tons,  0.55,  0.53. 

Legs  17.2  tons;  back  stay 

18.8. 
7.8  tons,  6.5. 

CB  =  W    '^  ; 
Sin  C 

sm  C 
At   C    force    is    horizontal 

and=WV3 


231. 
232. 
233. 

234. 


242. 
246. 
251. 


-;  at  B  tan-i 


V3 


^        lo      vertical     ?»nd 

^W  V7 
2 
14.24  pounds. 
Force  15  parallel  to  CA. 
13  units  at  tan— 1  /V  with 

AB. 
2  ^2  P  parallel  to   CA  at 

distance  from  AC  equal 


W/ 


to^^-^AB. 

2 

i:  V3- 

Tension  = . 

2  ^l^  —  c-^\\\^Q 
Reaction     115.5     pounds; 

tension  57.7. 
6d  000  pounds  close  to  the 

tower ;     47  000    in    the 

middle. 


ANSWERS. 


137 


256.  4  inches  from  end. 

257.  9.47  inches  from  end. 

258.  10  pounds. 

259.  9  inches  from   middle;   18 

pounds. 

260.  3  feet. 

262.  148I  pounds. 

263.  5  inches  from  middle, 

264.  3f  feet  from  B. 

265.  81  pounds. 

268.  1 06 1  pounds. 

269.  100  pounds,  50. 

270.  3    pounds ;    \    inch    from 

middle. 

271.  Its   ends   7    inches   and   3 

from  the  pegs. 

272.  4itons,  3f. 
274.    1 1  pounds. 

276.  6i  pounds. 

277.  89.143  pounds;  45.143. 

278.  120  pounds. 

279.  6  inches  from  end. 

280.  9.5  pounds. 

283.  4  feet  from  one  man. 

284.  1 1  inches,  6. 

287.  15  V5  pounds. 

288.  5  feet  from  end. 

289.  I    pound     at     distance    5 

feet. 

290.  540. 

291.  280  pounds,  392  ;   i  foot. 

292.  On   line    bisecting  vertical 

angle,  -|  from  vertex. 

9  3  ^^ 

the    sides,   if    each    side 

=  2  a. 

294.  — —-  a,  ^— ^  a,  — ^~  a 
II  II  II 

from  sides;   outside  the 

triangle       at       distance 

6V3^  Z^Z  ^  2  V3  ^ 
tty  a, a. 

5  5  5 

296.  Any  point  of  line  parallel 
to  CD  passing  through 
X  which  is  in  BC  pro- 
duced so  that  CX=  2  BC. 


298.  At   X  in  BD  produced  so 

that2DX  =  BD. 

299.  5   units  acting   parallel  to 

BD,  cutting  BC  produced 
in  X  so  that  4CX=:BC. 
305.    At  point  15  and   16  inches 
from  adjacent  sides. 

307.  200  pounds,  220. 

308.  A    parallel    force,    distant 

5  units. 

309.  2\  feet  from  rim. 

313.  If  D  be  the  middle  point 
of  BC,  R  is  represented 
in  magnitude  by  2  AD, 
and  acts  through  X  par- 
allel to  DA.  X  being  in 
DC  or  DB  so  that  DX 
_BC 
~~8"" 

315.    He  loses  i  pound. 

317.  I  inch. 

318.  85.9  pounds. 

321.  Each  equal  15  pounds. 

322.  32!  pounds. 

323.  2  700  pounds. 

325.  11.25  pounds,  31.25. 

326.  Length  of  stick  from  nail 

•    to       wall  =  V3       fs^- 
Pressure     =8-^3      and 

8  v^9— I  ounces. 

327.  18  900  pounds. 

328.  35-3  feet. 

331.  15  000  pounds. 

332.  f     of     length     from     end 

where     pressure     is      4 

pounds. 
32%  feet  below. 
7    ounces;     C.    of    G.    9)- 

inches  from  A. 
21   pounds  at  A,  9  pounds 

at  B  ;  additional  pressure 

10  pounds 


335. 
336. 

337. 


339. 


I  inch  from  AC,  i^ 

from  AB.     • 
1  W  ;  I  W. 

3COS— - — 


:hes 


340 

348.    2  cos  e 


U8 


ANSWERS. 


350 
331. 
353. 
354. 

355. 

357. 

358. 
359. 
362. 
365. 
366. 
871. 
372. 
375. 

376. 


379. 
380. 

381. 

382. 
384. 
385. 
386. 
387. 
388. 
390. 

391. 

393. 
394. 
395. 
396. 
398. 
400. 
401. 
403. 
404. 
406. 
407. 


9V3. 

y3jy  pound. 

45  degrees  to  the  vertical. 
It  makes  angle  X  with  nor- 
mal. 
373  pounds. 

0.35  ;  0.26. 
200  pounds. 

7. 

TT- 

50  pounds,  50  \[t^. 

IX  =^ ;  Inclination  =  tan— 1 1. 

423.3  foot-pounds. 

60  degrees. 

7.56  tons,  396. 

(I  +  /U.2)  sin  a 

r    =  radius. 

W  =  weight  of  wheel. 
47  feet, 
50  feet. 

1  920  pounds. 
65  pounds. 

2  800  pounds. 
164  pounds. 

3h 

246  pounds,  27.2. 

504.12  pounds. 

W 

—  =  0.952  or  1.05. 

8  inches. 
970  pounds. 

8^ 
222 

1.92  horse-power. 
155  thermal  units. 
4.13  horse-power. 
0.51  horsepower. 
0.44. 
35  feet. 

2  k  miles ;  from  A  3  miles, 
"  B  I  mile. 


408. 
409. 
410. 
411. 
412. 

414. 

416. 
417. 
418. 
419. 

420. 
421. 


423. 
424. 

425. 
426. 
431. 

432. 
433. 

434. 
435. 
436. 
437. 
438. 
439. 
440. 
441. 


442. 
444. 
445. 
449. 
452. 
454. 
455. 

457. 


30  miles  per  hour. 

2  640  feet. 

99  feet. 

150  feet,  200. 

65.5   miles  per  hour;  0.27 

miles. 
6    seconds ;     112   feet   per 

second. 

y/a^  feet  per  second. 
784  feet. 
231  feet. 

306.05  feet;  68.01  feet  per 
second. 

4  080  feet. 
■-,  ji 

- —  from  ground  ;   in  time 

352  feet. 
6  and  6. 

^^  V3  ;    u^ 

2  2 

17-6  feet. 
"^    feet  per  second. 

8.94  miles  per  hour. 

N.  W.       6  \/2   miles    per 

hour. 
10  miles  per  hour. 
I3j"ymiles  per  hour,  27 j\. 
24  7r. 

2.14  miles. 
ii 

5  seconds. 

2  feet  per  second. 

(i)  5  seconds;  (2)  200  feet; 

(3)  5  seconds;  (4)  80 feet 

per  second. 

6  feet  3  inches. 
3iyfV  tons  weight. 

4|  tons. 
3y'^  pounds. 
I  second. 
8  feet  per  second. 
{a)  -y^feet ;  (h)  70  pounds, 

140,  i86f. 
16  feet  per  second. 


AiVSlVEKS. 


139 


,    40 

nward  from 


458.    6  547.2    feet ;      29.5    miles 

per  hour. 
7.25  feet 
2^  pounds,  T,\. 
50    feet    per    second 

degrees  downward 

horizontal. 

16  VS    feet    per    second; 

100  feet. 
15  degrees  or  75  degrees. 
10  seconds ;  i  440  V3  feet. 
42  ^y2  feet. 
470  feet. 
7x52  n^iJes. 
144  feet 


459. 
462. 
463. 


465. 


466. 
467. 
468. 
469. 
471. 
475.    .^^  -_- 

477.  0.3  ton  weight. 

478.  60  miles  per  hour. 

479.  " 
480. 
482. 


— "-"  1^"-' 

2.83  pounds. 

10. 

i5y\  revolutions. 


483.  tan-i^igV^. 

484.  32.19. 

485.  3  666|  feet. 

487.  60  feet  per  second. 

488.  I  li  units. 

489.  3    inches;     160    tons   plus 

weight  of  mass  ;   10  000. 

491.  33  f  feet. 

492.  I  250  tons. 

493.  ID  feet  per  second. 

494.  6.9  feet. 

496.  —  I  feet  per  second,  2. 

497.  1 3. 1  miles  per  hour. 

498.  496.8  units. 

499.  8y  feet  per  second. 

500.  A    returns    at    5    feet    per 

second.  B  moves  at  45 
degrees  with  its  course 
and  velocity  of  10  \/2feet 
per  second. 


ALPHABETICAL     CLASSIFICATION 
OF    PROBLEMS. 


Acceleration,  body  moved,  446,  447. 
body  on    table    moved    by  .hanging 

body,  450. 
elevator    descending  and    ascending, 
454,455- 
Accumulator,  horse-power  of,  107. 
Action  of   rudder   in  counteracting  lee- 
way, 241. 
of  wind  on  sails,  239. 
Activity,  or  power,  of  dynamo,  58. 

engine  raising  water  from  mine,  4. 
Advantage,  mechanical,  of  differential 
screw,  35. 
four  pulleys  in  two  blocks,  383. 
wHeel  and  axle,  168. 
Aim  to  hit  running  deer,  426. 

vertical  angle  of,  to  hit  target,  466. 
Algebraic  sum   of    moments  of  forces, 

290. 
Amoskeag     Manufacturing    Company, 

h^rse-power  of  canal,  103. 
Anchor,  man-power  to  lift,  44. 

raised  by  capstan ,  length  of  spokes, 
266. 
Angle  of    inclination,   thin    suspended 
plate  weighted  at  corner,  343. 
two  forces,  173. 
An^le  of  friction,  354. 

weight  on  table  just  slides,  356. 
Angular  velocity  of  fly-wheel,  436. 
Apparent  direction  of  wind  from  mov- 
ing train,  433. 
and  true  velocity  of  wind  from  mov- 
ing steamer,  434. 
Arms  of  balance  unequal,  314. 

weigliingin  alternate  pans,  315. 
Axle  frictiDn ,  simple  pulley,  389, 390,391 . 
horizontal,  heat   generated  and  heat 

lost  in  friction,  400,  401. 
wheel  and,  weight  raised  by,  37. 

Balance,  bar  weighted  one  end,  258. 
incorrect  when  loaded,  316. 
position  of  weights  on   lever  to  bal- 
ance, 264. 
unequal  arms,  314. 

weighing  in  alternate  pans,  315. 
Ball   held   on   incline  by  string,  forces 
acting,  246. 


Ball  in  box,  pressure  on  sides,  212. 
on  end  of  string  swmging,   velocity, 

158. 
pierces  shield,  loss  of  energy,  147. 
projected  up  incline,  lime,   distance, 

and  velocity,  441. 
rising,  meets  one  falling  when    and 

where,  421. 
Balloon,  bullet  strikes, relative  velocity. 

429- 
height   of,  by  dropping  stone  from, 

419,  420. 
held   by   inclined   rope,   tension   and 

wind  pressure,  190. 
moving,  weight  of   body  in  balloon, 

453-. 
Ear  carried  by  two  men,  weight  borne 

by  eacii,  26). 
three  forces  acting  on,  resultant,  289, 
weighted    both   ends,    fulcrum,    256, 
257,  259.  263. 

one  end,  balance,  258. 
weighted  one  end,  weight  to  balance, 

276. 
Barge,  coal  hoisted  from,  horse-power 

required,  79. 
Bead  on  circular  wire,  172. 
Beam  against  vertical   wall  just  slips 

friction  coefficient,  381. 
hanging  by  one  end,  foot-pounds  to 

raise.  27. 
held   against    smooth    wall  by  rope, 

tansion  and  pressure,  201. 
held    against   wall    and  ground    by 

string,  tension   and  pressure,  319, 

322.  ■ 

in   hemispherical  bowl,  rests  against 

wall,  equilibrium,  245. 
inclined,  held  by  wall  and  string,  ten- 
sion, 320. 
rests  on  ground  and  incline,  reactions, 

324- 
rests   on   inclined  planes,  pressures, 

214. 
supported    by    inclined   chains,  ten- 
sions, 307. 
supported   by  two   strings,  tensions, 

191. 
Belt,  difference  of  tensions  in  sic'.es,  63. 


142 


MECHANICS-PROBLEMS. 


Belt,  driving,  width  of,  6^^. 
maximum  pull  on,  394. 
least  speed  of,  for  given  horse-power, 

65. 
on  pulley,  power  transmitted,  392. 
width   of,    to   transmit  given    horse- 
power, 393. 
Bicyclist    rides   against   wind,  force  of 
whid,  127. 
rides   up   hill,   work   per   minute  re- 
quired, J  26. 
overcomes  lesistance,  128. 
Bitts,  hawser  around,  to  stop  surging, 

^87. 
Blacksmith's  helper  swings  sledge,  rate 

of  work.  165. 
Block  dragged  on  table,  reaction,  353. 
on  incline  just  slides,  friction,  358. 
turns  over  before  slipping,  friction, 

369- 
rough  floor,  slip  or  turns  over  when 
pulled,  368. 
of  stone  carried  by  two  men,  283. 
Boat  crossing  river,  place  of  landing, 
428. 
crossing  river,  velocity    and  course, 

427. 
eight-oared,  propelling  force  of,  262. 
on  davit,  forces  at  loot-step  and  col- 
lar, 285. 
pulled   along  canal  by  men  on  each 

bank,  effective  pull,  192. 
towed  by  inclined  rope,  effective  pull, 

'93; 
velocity  moves  away,  man  jumps  in, 

493- 
Body  falling  down  incline,  170. 
on  rough  circular  arc,  160. 
held  on  incline  by  string  over  pulley, 

238. 
in  moving  balloon,  weight  of,  41;. 
lifted     by    two     persons,    resultant, 

177. 
moved,  acceleration,  446,  447. 
from  rest,  distance  traveled,  410. 
force  required,  448. 
work  required,  166. 
moving  horizontally  against  friction, 
velocity,  457. 
overtakes   another    body,  velocity 
and  energy, 499. 
moving,  velocity  under  constant  ac- 
celeration, 405. 
on  "rough  horizontal  plane,  least  pull 
to  drag,  365. 
incline,  force  to  pull  it  up,  364. 
force  to  support,  362. 
held  by  weight  on  string  over  pul- 
ley, 366. 
on  table  moved  by  hanging  body,  ten- 
sion in  thread,  449,  450. 
projected  down  incline,  distance,  464. 
projected  up  incline,  time,  distance, 
and  velocity,  441. 
obliquely  upwards,  magnitude  and 
direction  of  velocity,  463. 


Body  rests  on  rough  incline,  parallel  and 
perpendicular  forces,  208. 
slides  down   incline,  components   of 
velocity,  425. 
distance,  423. 
space  passed  over,  422. 
velocity  and  height,  46^. 
Bodies,  falling,  distance  between  them, 
4'3- 
moving,  meet  at  right  angles,  impact, 
velocities,  and  directions,  500. 
Boiler  capacity  for  fire-pump,  1 10. 
horse-power  for  fire-pump,  108. 
supported  by  tackles,  forces,  217. 
Bolt,  tension  in,  when  tightened  up  by 

wrench,  382. 
Boom  and  tackle,  stresses  in,  187,  189. 

228. 
Boston,  Mass.,  horse-power  of  tide  at, 
132. 
Thanksgiving    Fire  at,  coal  required 
for  engine,  161. 
Bowl,  hemispherical,    bar    in,   equilib- 
rium, 24). 
beam  in,  rests    against  wall,  equilib- 
rium, 245. 
rod  partly  in,  reactions,  250. 
weight  held  in,  by  string  from  rim,  248. 
Box  with  cover  open,  center  of  gravity, 

342. 
Box-machine  table,   moved   by  falling 

weight,  451. 
Brake,  lever  in  hand-wheel  of,  309. 
friction,  testing  power  of  engine,  395, 
396. 
water  motor,  39-. 
Westinghouse,  resistance  of,  131. 
Brakes  stop  train,  coefficient  of  friction, 

.  438. 
Bridge,  loaded,  pressures  in   supports, 
272. 
stresses  in,  254. 

suspension,  forces  in  cables,  251. 
truss  stresses,  255. 
stress  in  post,  323. 
Bucket    dropped    into    well,   depth    of 

well,  418. 
Buffer  stops  car,  energy  exerted,  23. 
Bullet  fired,  height  of  fall  for  given  ve- 
locity, 151. 
initial  velocity,  495. 
powder  pressure,  149. 
range  and  height,  470. 
stopped    by    sand-bank,  distance    of 

penetration,  150. 
strikes    balloon,  relative    velocities, 

429. 
vertical  angle  of  aim,  466. 

Cable  car  "  runs  wild,"  distance  passed 

over,  423. 
Canal    boat,  advantage    of    long  rope 
over  short  in  pulling,  179. 
pulled  by  mules  on  each  bank,  effec- 
tive pull,  192. 
lock,  horse-power  to  pump  out,  114. 


ALPHA  BE  TICAL    CLASSIFICA  TION. 


143 


Canal,     Manchester,     theoretic    horse- 
power of,  103. 
Merrimack,  number  of  looms  it  will 

drive,  104. 
Holyoke,  number  of  paper  machines 
it  will  drive,  118. 
Candle-power  per  horse-power,  59. 
Cannon,  distance  of  recoil,  154. 
recoils,  distance    moved  up   incline, 
442. 
Cannon-ball,  fired    from    hill,   time   to 
strike  sea,  472. 
greatest  range  of,  471. 
Capacity  of  boiler  for  fire-pump,  no. 
Capstan  raises  anchor,  length  of  spokes, 
266. 
weight  on  chain,  work  done,  11. 
Car-dumper  at  Carnegie  Works,  stresses 

in  supporting  frame,  222. 
Car,  electric,  current  to  propel,  6r,  134, 

strikes  buffer,  energy  exerted,  23. 
Cars  start  from  rest  down   incline,  dis- 
tance and  velocity,  458. 
Carriage  and  man,  relative  velocity,  432. 
passed  by  train,  relative  velocity,  430. 
Cart  drawn  by  horse,  pull  necessary,  29. 
Center   of   forces    acting   at  corners   of 
equilateral  triangle,  293,  294. 
in  hexagon, 300. 
in  square,  298. 
Center  of  gravity,  box  with  cover  open, 
.342.       .  .^ 

circular  disk  and  wire,  349. 

punched, 341,  344. 
curtain -rod,  334. 
hemisphere  and  cone,  347,  350. 
right    triangle  weiglited    at    corners, 

339- 
rod  on  two  supports,  336. 
solid  cylinder,  within  hollow  one,  346. 
square  and  equilateral  triangle,  338. 
T-shaped  rods,  333. 
T-section,  345. 
masses  above  and   below   horizontal 

line,  33 "5 ■ 
triangular  mass  of  rocks  raised   how 

far,  285. 
Z-section,  345. 
Chalk,  cylindrical   shaft  sunk  in;  work 

done, 14. 
Chain  and  weight  hoisted,  work  done, 
II. 
holds  trap-door  open,  forces  in  chain 

and  hinges,  330. 
wound  up,  work  done,  10. 
Chains,  two    inclined,  support    beam, 

tensions,  307. 
Charles  River,  Horse-power  of,  98. 
Chimney   of    Clark    Thread  Company, 
foot-pounds  in  construction,  9. 
pulled  over  by  rope,  point  of  attach- 
ment of  rope, 328. 
Circular  arc,  body  falling  along,  160. 

wire,  bead  on,  172. 
Coal,  amount  burned  and  efficiency  of 
steam-engine,  91. 


Coal,  amount  burned  and  hone-power 
of  steam-engine,  92. 
raised  by  given  horse-power,  124, 
required  for  engine  at  "  Thanksgiving 

Fire,"  Boston,  162. 
hoisted    from  barge,  horse-power   of 
engine,  79,  90. 
six-masted    schooner,    horse-power 
of  engines,  78. 
car  down   grade,    resistance  in  stop- 
ping, 80. 
wagon,  pressure  to  lift,  331. 
Coal-pit,  engine  draws  cage    from,  ten- 
sion in  rope,  461. 
Coal  supply  for  pumping  engine,  113. 
Coefficient   of    friction,     beam    against 
wall  just  slips,  381. 
block   on  incline  held  by  weight  on 
string  over  pulley,  366. 
just  slides,  358. 

turns  over  before  slipping,  369. 
ladder  against  rough  wall  and  ground 

just  slips,  378. 
Morin's  experiments,  350-351. 
stone  just  slides  down  hill,  357. 
train  stopped  by  brakes,  438. 
weight  moved  on  table,  352. 
about  to  move  on  table,  356. 
Collar  and  foot-siep  of  davit,  forces  at, 

285. 
Columbia,  U.  S.  warship,  resistance  to 

passage,  136. 
Component  forces    on    incline,    weight 
supported  by,  207. 
two  horizontal,  resultant  of,  175. 
Components,  horizontal  and  vertical,  of 
inclined  force,  178. 
of  velocity  along  diagonal  of  square, 
424. 
body  slides  down  incline,  4^5. 
Compressive  forces   in    legs    of  tripod, 

223. 
Cone  and  hemisphere,  center  of  gravity, 

347,350-    ,. 
on  incline,  slip  or  turn  over,  367. 
Conical   pendulum,   tension    in    string, 

479,  481. 
Connecting-rod,      thrust      in,    moment 

about  crank-pin,  329. 
Cord  over  pulley  weighted  at  both  ends, 

tension,  460. 
CorHss  engine,  horse-power  of,  93. 

Pacific     Mills,    Lawrence,    Mass., 
89. 
Cotter,  force   exerted    by,  and  pull   to 

withdraw,  373. 
Couple  and  force,  resultant  of,  308,  311. 
equivalent,  moment  of,  312. 
moment  of,  310. 
three  forces  equivalent  to,  313. 
to   turn    wheel   resting   between   two 
inclines,  376. 
Course  and   velocity   of  boat    crossing 

river,  427. 
Crane,  forces  in  jib  and  stays,  227. 
steam,  work  done  and  wasted,  77. 


144 


ME  CHANICS-PKOBL  EMS. 


Crane,  traveling,  horse- power  of  engine 
to  drive,  87. 

Cricket  ball  struck,  impulse,  488. 

Cross-bow,  energy  in,  21. 

Cubic  feet  to  descend  in  water-fall,  102. 

Current     output   of  turbine-driven  dy- 
namo, 60. 
to  propel  electric  car,  134, 

Cutting    tool,   horse-power    expended, 
125. 

Cylinder,    locomotive,    steam    pressure 
for  hDrse-power,  71. 
steam   engine,    pressure     per    square 
inch  for  given  horse-power,  85. 

Davit,  forces   at  foot-step  and  bearing, 

285. 
Deer  running,  aim  to  hit,  426. 
Deflictian  of  rope,  weight  between  two 

pulleys,  41. 
Depth   of   well   by  dropping   in  stone, 

415.  417,  418. 
Derrick,  stresses  in  tackle  and  boom, 

187,  t88. 
Diagonal     of     square,   components    of 
velocity  along,  424. 
and    sides    of   square,     forces   acting 
along,  res  iltant,  232,  233. 
Differential  pulley,  36,  37. 
screw,  mechanical  advantage,  35. 
wheel  and  axle,  37,  168. 
Dipper  dred:5e,  forces  acting  in,  228. 
Direction,  apparent,  of  wind  from  mov- 
ing train,  433- 
of  revction  of  rough  plane,  354. 
Directions  and    velocities  after  impact, 

bodies  meet  at  right  angles,  500. 
Discharge  of  fir^-engine,  iii. 

kinetic  energy  of,  116. 
Disk,  circular,  and  wire,  center  of  grav- 
ity, 349- 
punched,  center  of  gravity,  341,  344- 
Distance  ahead  of  running  deer  to  aim, 
426. 
and    velocity,   cars    start  from    rest 

down  incline,  458. 
apart   of    twj    trains   moving  toward 

each  other,  407. 
between  falling  bodies,  413. 
body  projected  down   rough  incline, 

464. 
cannon   moves    up  incline  by  recoil- 
ing, 442- 
elevator  moves  after  steam  is  shut  off, 

142. 
man  ascends  ladder  against  a  rough 

wall,  379,  380. 
of  recoil  of  cannon,  154,  494. 
penetration,    shot  stopped    by   sand- 
bank, 150. 
shot  penetrates  a  target,  148. 
traveled  before  stopping,  freight  car, 

'37- 

locomotive  and  tram,  6g. 

by  body  moved  from  rest   by  con- 
stant force,  410. 


Distance  travelled  before  moving  train 
brouglit  to  rest,  409. 
stone  moving  with  decreasing  velo- 
city, 406. 
stone  skimming  on  ice,  411. 
by  train  retarded  by  resistance,  437. 
Draw-bar   pull,    train   ascending   grade 

with  acceleration,  445. 
Driving  belt,  difference  of  tensions  in 
sides  of,  63. 
maximum  pull  on,  394. 
width  of,  66. 
pulley,  speed  of,  64 
Drum  and  gearing  raising  weight,  38. 

of  steam  windlass,  rope  around,  384, 
Dump-car    pulled    by    inclined    chain, 

effective  pull,  206. 
Dynamo  driven  by  turbine,  current  out- 
put, 60. 
kilowatts,  58,  62. 

shaft,  horse-power  lost  in  heat  genera- 
tion, 402. 
Dynamometer     readings.     Part   of  cir- 
cumference incircled,  396. 

Earth  drawn  out  of  well  by  horse,  work 
done,  13. 
shaft  sunk  in,  14. 
Effective  force  on  locomotive  with  in- 
creasing speed, 443. 
Effective  pull,  boat  tawed   by  inclined 
rope, 193. 
canal   boat  pulled   by  mnn   on   each 

bank,  192. 
dump-car  drawn  by  inclined  chain ,  206. 
Effective  pressure,   oars   on   row-boat, 
26r. 
wind  on  sails,  240. 
Effective  work  in  pulling  tram-car,  43. 
Efficiency  of  engine  for  coal  burned,  91. 

pump,  117. 
Eight-oared  boat,  propelling  force,  262. 
Elastic  string,  work  done  in  stretching, 

169. 
Electric  car,  current  to  propel,  61,  134. 
kinetic  energy  of,  146. 
current  for  hoisting  apparatus,  18. 

from  water-power,  60. 
lamps,  horse-power  for,  159. 
Elevator  descending,  acceleration,  454, 

455- 

Elevator   lifted,   distance   moved    after 
steam  is  shut  off,  142. 
distance  passed  over  for  time  and  ac- 
celeration, 455. 

Empire  State  Express,  locomotive,  375. 

Emptying  tank,  time   required  for  two 
men,  65. 

End  supports,  pressure  on,  due  to  weight 
hung  between, 337. 

Energy   and    range   of    projectile   fired 
from  hill,  473. 
velocity,  one  body  overtakes  another, 

499. 
consumed    by    man    walking   up   in- 
cline, 366. 


A  L  PHA  BE  TIC  A  L    CL  A  SSIFICA  TION. 


145 


Energy  expended  by  body  sliding  down 

incline,  30. 
in  cross-bow,  21. 

balls,  171. 
kinetic,  of  discharge,  n6. 

one  ball  overtaking  another,  498. 

projectile,  133. 

tram-car,  145. 
lost,  ball  pierces  shield,  147. 
pendulum  bob,  157. 
potential  ot  water-fall,  12. 
projectiles  from  rapid-fire  gun,  133. 
stream  of  water  flowing,  12.. 
Engine,  at  water-works,  horse-power  of, 

112. 
draws  cage  from  coal-pit,  tension  in 

rope,  4b I. 
efficiency  for  coal  burned,  91. 
force  exerted  by,  to  move  train,  444. 
horse-power  of,  81,  86,  87,  93,  94,  95, 

96.97-  ,  ,     . 

horse-power  and  revolutions,  88. 
horse-power  for  coal  burned,  92. 
horse-power  to  raise  water,  53. 
power  tested   by  friction   brake,  395, 

396- 
rounding  curve,  pressure  on  rails,  478. 
steam  pressure  for  given  horse-power, 

steam  pressure  on  piston  guides,  194. 

speed  maintained,  57. 

water  pumped  from  mine,  4. 

work  done  in  overcoming   resistance 

of  train,  2. 
working  twenty  forge  hammers,  horse- 
power of,  50. 
Equilateral  triangle  and  square,  center 

of  gravity,  338. 
forces  acting  at  corners,  center  of,  293, 

294.       . 
forces  acting  at  corners,  resultant,  292. 
Equilibrium,  bead  on  wire,    sustaining 

weight,  172. 
bar  in  hemispherical  bowl,  249. 
beam    in    hernispherical     bowl    rests 

against  wall,  245. 
heavy  ring  on  cord,  243. 
rod    supported    by  smooth    pin   and 

string,  236. 
rod  supported  by  smooth   wall  and 

fixed  point,  200. 
sinker  in  running  water,  235. 
stable  and  unstable,  347. 
stick  rests  on  nail  and  wall,  326. 
weight  sliding  on  thread  fixed  to  rod 

free  to  turn,  348. 
Equivalent  couple,  moment  of,  312. 

Factory,  pump  for  fire  protection,  no. 
Fall, height  of,  forgiven  velocity  of  bul- 
let, 151. 
of  water,  foot-pounds  per  minute,  12. 
Fall  River  Cotton  Mills,  86. 
Falling  bodies,  distance  between  them, 
413- 
body  drags  mass  along  table,  171. 


Falling  body  drags  on  circular  arc,  160. 
power  developed  and  friction  over- 
come, 42. 
weight  works  pump,  gallons  of  water 
raised,  26. 
moves  weight  on  table,  time,  452. 
Fire-engine,  coal  required  for,  162. 
useful  works  done  by,  i6r. 
discharge  and  horse-power,  iii. 
Fire-pump,  boiler  capacity  for,  no. 
boiler  horse-power  for,  108. 
Underwriter,  work  done  by,  log. 
Floor,  rough,  block  slip    or  turn  over 
when  pulled,  368. 
supports,  pressure  on,  due  to  metal- 
planer,  304. 
Fly-wheel,  angular  velocity  of,  436. 
revolutions  before  stopping,  159,  163, 
164. 
Foot-bridge,  stress  in  post  of,  323. 
Foot-pounds      developed     by     falling 
weight,  42. 
steam  in  cylinder,  94. 
expended  by  horse  lifting  earth  from 
trench, i6. 
by  body,  sliding  down   incline,    17, 

30. 
on  train  resistance,  i. 
on  tram-car,  43. 
by  men  lifting  weight,  5. 
per  minute  of  water-fall,  13. 
pumping-engine  lifts  water  from  mine, 
4,  24. 
pumps  water  from  tank,  25. 
raising  material  from  well,  15. 
to  lift  material  for  monument,  9. 
to  lower  water  level  in  well,  8. 
to  punch  hole  in  wrought  iron,  3. 
to  raise  beam  hanging  by  one  end,  27. 
to  raise  material  from  depth,  45. 
to  sink  shaft  through  chalk,  14. 
wasted  on  friction  in  steam  crane,  77. 
Foot-step  and  collar  of  davit,  forces  at, 
285. 
bearing,  horse-power  lost,  403,  404. 
Force  and  couple,  resultant  of,  3  8,311. 
at  end  of  lever  inclined,  318. 
at  one  end  of  lever,  275. 
exerted  by  cotter,  and  pull  to  with- 
draw, 373. 
exerted  by  engine  to  move  train,  444. 
exerted   by   horse  continuously  dur- 
ing day,  7. 
walking  around  circle,  31. 
exerted  on  buffers  in  stopping  car,  23. 
lifts    triangle   by   corner,  pressure  at 

base,  340. 
necessary  to  raise  a  weight  with  pul- 
ley, 28. 
of  hammer  driving  nail,  155,  156. 
of  hammer  of  pile-driver,  ig,  20. 
of  powder,  shot  fired  from  gun,  492. 
to  balance  two  forces  acting  on  rod, 
288. 
I        control    weight     lowered     by     rope 
I  around  spar,  386. 


146 


MECHANICS-PROBLEMS. 


Force  and  couple,  move  body  from  rest, 

448.     - 
overturn  trapezoidal  wall,  327. 
pull  wheel  over  stone,  211. 
stop  train  in  given  distance,  22. 
support  body  on  rough  incline,  362. 
Forces  acting,  along  sides  of  hexagon, 

moments,  301. 
square,  resultant,  231,  234. 

sum  of  moments  about  point,  295. 
triangle,  sum  of  moments  about  base, 

294. 
two   sides    and  diagonal    of    square, 

resultant,  232,  233. 
ball  held  on  incline  by  string,  246. 
in  cables   of  suspension  foot-bridge, 
.    251- 

m  chain  and  hinges  of  trap-door,  330. 
in  dipper  dredge,  228. 
in  hexagon,  center  of,  300. 
in  square,  center  of,  298. 
in  square,  moments  of,  297. 
in  tackles  supporting  a  boiler,  217. 
on  hinged  rod  resting  on  peg,  237. 
weight  in  hemispherical  bowl  held  by 

string  from  rim,  248. 
Forces,  at  corners   of    equilateral    tri- 
angle, center  of,  293. 

resultant  of,  292. 

of  square,  force  to  balance,  305. 
resultant,  306. 
at  right  angles,  resultant  of,  174. 
component   on   incline,    weight   sup- 
ported, 207. 
compressive  in  tripod  legs,  223. 
four  at  a  point,  resultant,  182. 
in  jib  and  stays  of  crane,  227. 

tie  of  crane,  188. 
in  legs  and  stay  of  shears,  225,  226. 
inclined  at  an  angle,  17^. 
inclined,  horizontal  and  vertical  com- 
ponents, 178. 
on  three  sides  of  a  rectangle,  resul- 
tant, 299. 
parallel  and  perpendicular,  body  rests 

on  incline,  208. 
three  act  on  bar,  resultant,  287,  289. 
three  at  a  point,  resultant,  181 
three  equivalent  to  a  couple,  313. 
Forj^e  hammer,  horse-power  of,  50. 
Fortification  wall,  landing-place  of  pro- 
jectile fired  over, 474. 
Fourdinier    Taper    Machines  at    Hol- 

yoke, 118. 
Fragments  of  shell,  original  momentum, 

498. 
Freight      car      side-tracked,     distance 

moved  before  stopping,  137. 
Friction,  amount  of  heat  generated  by 

revolving  sliaft,  399,  400,  401 ,  402. 
angle  of, 3^4,  356. 
axle  of  pulley,  301. 
axle,    single    pulley,    pull    to     raise 

weight,  389,  390. 
between   wheels    and    rails,   pull    of 

locomotive,  375. 


Friction,  brake  testing,  power  of  engine, 
395.  396- 
power  of  water  motor,  397. 
coefficient,    train   stopped  by  brakes, 

438. 
coefficients    of    different   substances, 
.     ,3527353- 
mchnation    of    tennis-net  poles  due 

to,  370. 

shaft  bearings,  30,8,  399,  400,  401,  402. 

Fulcrum,  at  middle   of  bar,  distribution 

of  weights  to   bring,  280. 

balance  incorrect  when  loaded,  316. 

bar  weighted  both  ends,  256,  257,  259, 

263. 
pressure  on  lever  held  by  pin,  277. 
uniform  heavy,  lever,  279. 

g,  value  of,  at  London,  484. 
Gas  engine,  horse-power  of,  51. 
Gearing,  drum  and  weight  raised  by,  38. 
Grade,  speed  of  locomotive  up,  70. 
Graduations,   length  of,  for  pound    on 

steelyard,  317. 
Ground,   time   of  stone   projected   up- 
ward to  reach,  414. 
Guides,  piston,  pressure  on,  in  steam 

engine,  194. 
Gun  lifted   by  shears,  Maryland  Steel 
Company,  224. 
on  inchne,  recoil,  491. 
pull    to     drag    up    inclines    having 

friction,  414. 
rapid-fire,       projectiles     discharged, 

horse-power  expended,  133. 
shot  fired  from,  impulse,  493. 
recoil,  494. 

Hammer  drives   nail,    force   of    blow, 

155,  156.    . 
of  pile-driver,  force  of,  19,  20. 
Hammock-ropes,  pull  in,  19;. 
Hand-wheel  of  brake,  lever  in,  309. 
Hawser   around   bitts  to  stop   surging, 

387. 
Hay  scales,  weighing  half-load  at  time, 

273- 
Head  wind,  force  of,  against    bicycle 

rider,  127. 
Heat  generated   by  friction,  revolving 

shaft,  399,  400,  401 ,  402. 
Heavy  ring  on  cord,  equilibrium,  243. 
Height   and  greatest   range   of   bullet, 

470. 
Height  of  balloon   by  dropping  stone 

from,  419,  420. 
mountain  top  by  pendulum,  485. 
water  fail  required,  loi. 
Hemisphere  and  cone,  center  of  gravity, 

347.  350- 
Hexagon   center  of  forces  acting  in,  300. 
moment  of  forces  acting  along  sides, 

Higli-speed   engine,  horse-power  of,  81 » 

Hinged   rod   held    by    inclined    string, 

tension  and  thrust,  183,  185,  321. 


ALPHABETICAL    CLASSILVCA  TION. 


147 


Hinged  rod   supported   by  peg,  forces 
acting,  237. 
weighted,  equilibrium,  274. 
Hoisting  apparatus,  current  for,  18, 

with  pulley,  force  necessary,  28. 
Hole  punched  through  metal  plate,  3. 
Hollow  cylinder  containing    solid  one, 

center  of  gravity,  346. 
Holyoke,  Mass.,  Whiting  Paper  Mills, 

118. 
Hoop,  ring  held  on  by  string,  reactions, 

244. 
Horizontal  and  parallel  pulls  on  incline, 

and     vertical     components,    inclined 

force,  178. 
axle,  heat  generated  and  horse-power 

lost,  400,  401 . 
components,  resultant  of  two,  175. 
plane,   bail    projected    on,    velocity, 

438. 
pole  fixed  one  end,  breaking,  260. 
string  holding  weighted  rod,  tension 

and  thrust,  184. 
Horse  draws  cart,  pull  necessary,  29. 
draws  earth  out  of  well,  work  done, 

13. 
draws  load  up  incline,  pull  on  traces, 

359- 
lifts  earth  from  trench,   work  done, 

16. 
power  developed  by,  7. 
power  expended  by,  in  raising  weight, 

46. 
walking  round  a  circle,  force  exerted, 
31- 
Horse-power,  accumulator,  107. 
and  belt  speed,  64,  65. 
boiler  to  run  fire-pump,  108, 
Corliss  engine,  93. 
electric  lamps,  59. 
engine  to  drive  traveling  crane,  87. 
engine  to  unload  coal  from  six-mas- 
master,  78. 

expended  at  cutting-tool,  125. 
fire-engine,  11  (. 
forge  hammer,  50. 
from  indicator  cards,  82,  83. 
gas-engine,  51. 
indicator  of  diagram,  84. 
locomotive,  54,  55,  72. 
down  incline,  74. 
up  incline,  7;,  76. 
lost  in  foot-step  bearing,  403,  404. 
machine  discharging  projectiles,  133. 
man  carrying  weight.  47. 
man  swinging  hammer,  143. 
Niagara  Falls,  106. 
Niagara  turbines,  T05. 
of  engine  for  coal  burned,  92. 
of  Charles  River,  98. 
paper  machines,  Foudrinier,  n8. 
planing  machines,  49. 
pumping-engine  at  water-works,  112 
raising  coal,  52,  79,  90. 
raising  water,  53. 


Horse-power,  rope-drive,  68. 

steam-engine,  8i,  8j,  94,  95,  96,  97. 

steamship,  135,  140. 

stream  of  water,  60. 

theoretic  of  Manchester  canal,  103. 

tides,  132. 

to  lower  surface  of  lake,  120. 

overcome  friction  in  shaft,  400. 

pump  out  canal  lock,  114. 

raise  weight,  46,  47,  48. 

turn  loaded  shaft,  398. 

train  at  given  speed,  56,  138. 

wasted  in  heat  in  dynamo  shaft,  402. 

waterfall,  iig. 

water-wheel,  99,    100,  loi,  102,    103, 

104. 
width  of  belt  to  transmit  given,  393. 
windmill,  48. 

Ice,  push  to  move  stone  on,  351. 
stone  skimming  on,  distance  traveled, 
411. 
Ice-boat  started  from  rest,  velocity  and 

space,  412, 
Ice-cike   slides     down    chute,   vertical 

distance,  422. 
Impact,  velocity  after  two  balls  meet, 
496. 
after  two  trains  meet,  497. 
Impulse  cricket  ball  struck,  488. 

weight  falls  on  pile,  489. 
Inclination,  angle  of,  two  forces,  173. 
angle     of    thin      suspended     plate 
weighted  at  corner,  343. 
of  tennis  net   poles   due   to  friction, 

370. 
of  string  holding  pUimb-bob  in   car 
rounding  curve,  483. 
Incline   and   level   ground,  beam  rests 
on,  reactions,  324. 
ball  iield  on  by  string,  forces  acting, 

246. 
block  on,  just  slides,  friction,  358. 
turns  over  before  slipping,  friction 
coefficient,  369. 
body  held  on   by  string  over  jjulley, 
238. 
projected  down,  distance,  464. 
projected  up,  time,  distance,  velo- 
city, 441. 
rests   on,  parallel  and  perpendicu- 
lar forces,  208. 
slides   down,   components  of  velo- 
city, 425. 
space  passed  over,  422,  423. 
velocity  and  height,  465. 
distance  moved  up,  by  cannon  recoil- 
ing, 442. 
horizontal  and  parallel  pulls  on,  205. 
horse  draws   load  up,  pull  on  traces, 

359-     „  .  , 

man   wralking  up,   energy  consumed, 

360. 
rod  resting  on,  reactions,  203. 
rough,  bodv  held  by  weight  on  string 
over  piilley,  366. 


148 


ME  CI/ A  NICS-PKOBLEMS. 


Incline,  rough,  heavy  cone,  slip  or  turn 
over,  367. 
sphere    hpld   on    by   string,   point  of 

attachment  of  string,  247. 
with  friction,  pull  to  drag  gun  up,  361. 
Inclined  and  horizontal  ropes  support 
man,  pulls,  195. 
chain    dump-car  drawn    by  effective 

pull,  206. 
foi-ce  at  end  of  lever,  3 18. 
force,  horizontal  and  vertical  compo- 
nents, 178. 
plane,  body  falling  down,  170. 

horse-power    of  locomotive   down, 

74. 
horse-power   of  locomotive  up,  75, 

76. 
work  done  by  body  sliding  down, 
17,  30. 
planes,  beam  rests  on,  pressures,  214. 
string  over  pulley  just  holds  bodies 

on, 363. 
wheel    resting  between,  couple  to 
turn,  376. 
planks    supporting    weights    held  by 

string  over  pulley,  210. 
rope,  boat  towed  by,  effective  pull, 

193' 
holding  balloon,  tension  and  wind 
pressure,  190. 
string    holding    hinged    rod,  tension 

and  thrust,  183,  185. 
tramway,  weight  of  trucks  on,  209. 
Indicated  horse-power,  from  cards,  82, 
83,84, 
gas-engine,  51. 
steam-engine,  96. 
vessel,  140. 
Indicator  cards,  horse-power,  82,  83,  84. 
Initial  velocity,  bullet  fired  from  gun, 

495- 
stone  rising  meet?  one  falling,  416. 
train  ascending  grade  by  momentum, 

456. 
Iron  rail  carried  by  six  men,  282. 
sphere     between    wall    and     incline, 

pressures,  213. 

Jack-screw,  ratio  weight  to  power,  167. 

Jet  of  water  driving  motor,  62. 

Jib  and  stays  of  crane,  forces  in,  227. 

tie  of  crane,  forces  in,  188. 
Jointed  rods,  pressure  in,  when  loaded, 

221. 
Journal  friction,  398,  399,  400,  401,  402. 

Kilowatts  of  dynamo,  58.  62. 
Kinetic  energy  of  discharge,  116. 

balls,  171. 

electric  car,  146. 

projectile,  133,  152. 

tram-car,  145. 
King-post  truss,  stresses  in,  255. 

Ladder  against  rough  wall  and  ground 
slips,  friction  coefficient,  378. 


Ladder,  slipping  position,  377. 

distance  man  can  ascend,  379,  380. 
Lake    Shore   and    Michigan    Southern 

R.R.,  pull  per  horse-power,  139. 
Lake  Sliore  train,  horse-power  of,  138. 

steam-pressure  used,  71. 
Lake  surface  lowered,  horse-power  re- 
quired, 120. 
Landing-place,  boat  crossing  river,  428. 
Lawrence,  Mass.,  Pacific  Mills,  8g. 
Leeway,  action  of  rudder  in  counteract- 
ing, 241. 
Length  of  pound  graduations  on  steel- 
yard, 317. 
Lever  held  by  pin,  pressure  on  pin  and 
fulcrum,  277. 
in  hand-wheel  of  brake,  309. 
loaded  one  end,  force  at  other,  275. 
position  of  weights  to  balance,  264. 
uniform  heavy,  fulcrum,  279. 
weight  of,  by  balancing,  278. 
weighted  one  end,   inclined  force  at 

other, 318. 
with  rope   over  pulley,   equilibrium, 
265. 
Like  parallel  forces,  resultant,  287. 
Load,  maximum,  on  two  inclined  wires, 

218. 
Loaded  shaft,  horse-power  to  turn,  398. 
Lock,  canal,  St.  Mary's    Falls,   horse- 
power to  pump  out,  114. 
Locomotive,  cylinder  pressure  for  given 
horse-power,  71. 
down  incline,  horse-power,  74. 
increasing   speed,  effective  force  act- 
ing, 443- 
maximum  speed  of,  57. 
on  level,  foot-pounds,  2. 
horse-power,  54,  55,  72. 
pull  of,  Empire  State  Express,  375. 
speed  in  given  distance  and  time,  69, 

73- 
up  rough  incline,  horse-power,  75,  76. 
London,  value  of  *';f  "  at,  484. 
Looms,  Pacific  Mills, Lawrence,  Mass., 
89.  _ 
Merrimack  Manufacturing  Company, 

lOJ. 

Loop  of  rope,  man  sitting  in,  horizon- 
tal and  inclined  pulls,  1915. 

Lowell,  Mass.,  Merrimack  Manufactur- 
ing Company,  104. 

Man    and    carriage,    relative    velocity, 

432. 
Man  ascends  ladder  against  rough  wall, 
379.  380. 
jumps  into  boat,  which  moves  away, 

velocity,  493. 
lifts  heavy  weight  in  descending  ele- 
vator, acceleration,  454. 
Man-power,  cawying'weight,  47. 
to  raise  anchor,  44. 
to  swing  hammer,  143. 
Man  pushes  on  spoke  or  wagon  body, 
relative  effects,  267. 


ALPHABETICAL    CLASSIFICATION. 


49 


Man  rides  against  wind  on  bicycle,  force 
of  wind,  127. 
up  liili   on   bicycle   work  required, 
126.  . 

rowing,  rate,  129. 
supported  by  horizontal  and  inclined 

ropes,  pulls,  195. 
walking  up  incline,  energy  consumed, 
360. 
Manchester  Canal,  horse-power  of,  103. 
Maryland  Steel    Co.,  shears  erected  at 

works  of,  224. 
Mass  dragged   along   table   by  falling 

weight,    171. 
Masses,    center    of   gravity    of    three, 
above   and  below   horizontal   line, 
335- 
Maximum   load   on  two   inclined  wires 
supporting  weight,  218. 
pull  on  driving-belt,  394. 
speed  maintained  by  engine,  57 
Mechanical     advantage    of  differential 
screw,  35. 
from  pulleys  in  two  blocks,  383. 
wheel  and  axle,  168. 
Men,  power  developed  by,  5,  6. 
pump  out  a  tank,  time  required,  121. 
six,  carry  iron  rail,  282. 
six,  with    rope    pull   over    chimney, 

328. 
twenty, lift  weight,  work  done  by  each, 

5- 
two,  carry  block  of  stone,  283. 
two,    carry     horizontal    bar,     weight 
borne  by  each,  269. 
Merrimac  canal,  section  of,  104. 
Merrimac  Manufacturing  Company,  104. 
Metal  on  metal,  friction  coefficient,  350 
to  351. 
on  oak,  friction  coefficient,  350  to  35f. 
Metal    planer,   pressure   on  floor  sup- 
ports due  to,  304. 
plate,  hole  punched  through,  3. 
removed  per  horse-power,  125. 
Mills,  fire-pump  for  protection  of,  108. 
Mine,  amount  of  water  pumped  from, in 
given  time,  115. 
water  pumped  from,  horse-power  re- 
quired, 123. 
work  expended,  4, 24. 
Moment  about  crank-pin,  thrust  in  con- 
necting-rod, 32(y. 
of  couple,  310. 
equivalent  couple,  312. 
forces  acting  along  sides  of  hexagon, 

301. 
forces  acting  in  square,  312. 
Moments   of  forces,  algebraic  sum   of, 
290. 
sum  of,  about  base,  forces  along  sides 
of  triangle,  295. 
point,  fo-'ces  along  sides  of  square, 
296. 
Momentum   of  train    ascending  grade, 
4^6. 
shell  before  bursting,  498. 


Monument,     lifting     materials,     work 

done, 9. 
Morin's  results,  coefficients  of  friction, 

350  to  351. 
Motor,  electric,  horse-power  of,  60. 
Mountain,  height  of,  by  pendulum,  485. 
Moving  body,  velocity  under  constant 
acceleration,  405. 
train,     apparent     direction     of    wind 

from,  433. 
train,  brought  to  rest,  distance  trav- 
eled, 409. 
train,  velocity  increasing  uniformly, 

408. 
weights  on  threads  over  pulley,  space 
passed  over,  459. 

tensions  in  threads,  462. 

Nail  and  smooth  wall  supporting  stick, 
equilibrium,  326. 
driven  by  hammer,  force  of  blow,  155, 

Nails,  two,  supporting  weighted    rod, 

pressure,  284. 
Niagara  Falls,  horse-power  of,  106. 

turbines,  actual  horse-power,  105. 
Nozzle  drives  water-wheel,  117. 

of  fire-pump,  discharge  through,  iii. 

kinetic  energy  of  discharge,  116. 

Oak  and  metal,  friction  coefficient,  350 

to  351. 
and  oak,  friction  coefficient,  350  to  351. 
Oar,  pressure   on     row-locks,   effective 

pressure,  261. 
Otto  gas-engine,  horse-power  of,  51. 

Pacific    Mills,    horse-power   of  engine, 

and  number  of  looms  driven,  89. 
"  Pan  American  "  dipper  dredge,  228. 
Paper   machines,   Fourdrinier,  at    Hol- 

yoke,  horse-power  for,  118. 
Parallel  and  perpendicular  forces,  body 
rests  on  incline,  208, 
tracks,   velocities   of  two    trains   on, 
435- 
Peg  supporting  hinged  rod,  forces  act- 
ing, 237. 
Pegs,  rod  rests     on    two,  equilibrium 
when  weighted,  270. 
position     to     give     equal  pressures, 
271. 
Pelton  water-wheel    tested   by   friction 

brake, 397. 
Pendulum  at    London,  value   of  "^," 

484. 
Pendulum,  height  of  mountain  figured 

from,  485. 
Pendulum  bob,  energy  of,  157. 
velocity  of,  158. 
conical,  tension  in  string,  479,  481. 
revolutions,  480. 
Persons,  two  lift  a  body,  resultant,  177. 
Picture  cord,  tension  in,  219. 
Pile-driver,  force  of  hammer,  19,  20. 
weight  falls  on ,  force  and  impulse ,  489. 


ISO 


ME  CHA  NICS-PR  OBL  EMS. 


Pin  and  string  supporting  rod,  equilib- 
rium, 237. 
holds  lever,  pressure,  277. 
Piston  guides,  pressure  on,  194. 
Pit,  amount  of  coal  raised  from,  by  given 
horse-power,  124. 
horse-power  to  raise  coal  from,  52. 
Pitch  of  screw  m  screw-press,  32. 
ratio  of  weight  to  power,  33. 
Plane,  rough,  direction  of  reaction,  354. 
waj^on   rests     on   incline,  equivalent 
forces,  202. 
Planer  rests  on  floor,  pressure  on  floor 

supports,  304. 
Planing-machine,  horse-power  for,  49. 

work  done  in  moving  table,  371. 
Planks,  -.veights   on   inchned,  held  by 

string  over  pulley,  210. 
Plate,  thin,  suspended,   weighted    one 

end    angle  of  inclination,  343. 
Platform,  triangular,  pressure  on   sup- 

pjrts,  303. 
Plumb-bob  hung  in  car  rounding  curve, 

inclination  of  string,  4S3. 
Point,  four  forces  at,  resultant,  182. 
rod   hung  by   two  strings  from,  ten- 
sions, 198. 
three  forces  at,  resultant,  181. 
three  strings  meet  at,  176. 
Poli  held  against  wall  by  string,  tension 
and  pressure,  325. 
horizontal,  fixed  one   end,  breaking, 
260. 
Potential  energy  of  water-fall,  12. 
Post  of  bridge  truss,  stresses  in,  323. 
rope  around,  tension  in  rope,  385. 
Pound  graduations,  length  of,  on  steel- 
yard, 317. 
Powder,  force  of,  shot  discharged  from 

gun, 492. 
Powder  pressure,  bullet  fired,  149. 
Power  applied  to  end  of  winch  handle 
to  raise  weight,  38. 
developed  by  falling  weight,  42. 
horse,  7. 
men,  6. 
expended  by  man  swinging  hammer, 

143. 
of  engine  tested  by   friction   brake, 

395.  396- 

of    water-wheel    tested    by    friction 
brake,  397. 

transmitted  by  belt,  392. 
rope,  68. 
Pressure,  beam  rests  on   two   inclined 
planes,  214. 

effective,  of  wind  on  sails,  240. 

in  supports  of  loaded  bridge,  272. 

in  two  jointed  rods  when  loaded,  221. 

iron    sphere     between   two    inclined 
planes,  214.  « 

ol     steam-engine    for    given    horse- 
power, 85. 

on  end  supports  due  to  weight  hung 
between,  337. 

on  fulcrum,  weighted  rod,  259. 


Pressure   on    head    of    smooth    screw, 

34-  . 
on    rails,  train   rounding  curve,   476, 

477.  478. 
on  sides  of  box  containing  ball,  212. 
on    supports    of    loaded    triangular 

platform,  302,  303. 
on   three  tacks  in  isosecles  triangle, 

197. 
on  two  nails  supporting  weighted  rod, 

2S4. 
on  two  supports,  uniform  rod,  332. 
to  lift  coal-wagon,  331. 
water  flowing  in  pipe  suddenly  shut 

off,  486.  487. 
Projectile   fired  from     hill,    range  and 

energy  of  blow,  473. 
fired  over  fortification  wall,  place  of 

landing,  474. 
kinetic  energy  of,  152. 
stopped  by  sand-bank,  resistance  and 

time,  49 ). 
Projectiles  from  rapid-fire  gun,  horse- 
power expended, 133. 
Propelling    force  of   eight-oared   boat, 

2'02. 

Pull,  average  on  tram-car,  43. 

effective,  boat  towed  by  inclined  rope, 
193- 

canal  boat  pulled  by  mules  on  each 
bank,  192. 

horse  exerts  during  day,  7. 

least  to  drag  body  on  rough  hor- 
izontal plane, 365. 

in  hammock  ropes,  i0. 

of  engine  per  horse-power  on  Lake 
Shore  and  Michigan  Southern  R. 
R.,  139- 

of  locomotive,  friction  between  wheels 
and  rails,  375. 

on  draw-bar,  train  ascending  grade 
with  Rcceleration,  445. 

on  driving-belt,  maximum,  394. 

on  traces,  horse  draws  load  up  in- 
cline, 359. 

on  train  at  constant  speed,  139. 

to  draw  cart,  29. 

to  drag  gun  up  incline  having  friction, 
361. 

to  raise  one  end  of  shafting  on  two> 
supports,  281. 

to  raise  weight  with  tackle,  39,  40. 

to  withdraw  cotter,  373. 

wagon  drawn  up  road,  204. 
Pulls,   horizontal   and    parallel    on   in- 
cline, 205. 

man  supported  by  inclined  and  hor- 
izontal rofJes,  195. 
Pulley,  differential,  36. 

driving,  speed,  64. 

force  to  hoist  witli,  28. 

rope  on,  fastened  to  lever,  equilib- 
rium, 26^. 

single,  with  axle  friction,  389,  390,  391. 

string  over,  holds  body  on  incline,. 
238. 


ALPHABETICAL    CLASSIFICA  T/OiV. 


151 


Pulley,  string  over,  just   holds  bodies 
on  inclined  planes,  363. 
tension   in   cord  transmitting    horse- 
power, 67. 
Pulleys,  four    in  two  blocks,  mechan- 
ical advantage,  383. 
weight    between   two,    deflection    of 
rope, 41. 
Pump,  boiler  capacity  for,  no. 
efficiency  of,  117. 

underwriter  fire,  work  done  by,  109. 
worked    by    falling  weight,    gallons 
lifted,  26. 
Pum ping-engine,  at  water  works,  horse- 
power of,  112. 
coal  supply  for,  113, 
horse-power  from  indicator  card,  83. 
raising  water,  foot-pounds  of  work,  4. 
Pumping  out  canal   lock,   horse-power 
required,  114. 
water    from   mine,    foot-pounds    ex- 
pended, 24. 
cistern,  foot-pounds  expended,  25. 
Pump-plunger,  wedge  used  to  set  up, 

374- 
Push  on   spoke  or  wagon   body,  rel- 
ative effect,  267. 
to  move  stone  on  ice,  351. 

Rail  carried  by  six  men,  282. 
Railroad,  car  stopped  by  buffers,  force 
exerted,  23. 
train  overcoming  resistance,  i,  130. 
vertical  height  equivalent  to  given 
velocity,  144. 
Rails,  pressure  on,  train  rounding  curve, 

476,  477.  478. 
Ram,  foot-pounds  to  raise,  141. 
Range,  and  energy  of  blow,  projectile 
fired  from  hill,  473. 
and  height  of  bullet,  470. 
greatest  of  cannon-ball,  471. 
Rapid-fire  gun,   projectiles  discharged, 

horse  power  expended,  133. 
Rate  at  which  man  rows,  129. 
of  heat  generation  in  loaded  shafts, 

399.. 
of  train   judged  by  angle  of  fall   of 

rain,  431- 
of    work,    sledge    swung    by    black- 
smith's helper,  165. 
Ratio,  work  to  power  in  screw-press,  33. 
Reaction,  beam  rests  on   level   ground 
and  incline,  324. 
block  dragged  on  table,  353. 
of  rough  plane,  direction  of,  354. 
of  weighted  rod  on  supports,  291. 
ring  held  on  hoop  by  string,  244. 
rod    partly    in    hemispherical    bowl, 

250. 
rod  resting  on  incline,  203. 
Recoil  of  cannon,  distance  of,  154. 
gun,  494. 

gun  up  incline,  491. 
Rectangle,   forces   on   three    sides,    re- 
sultant, 299. 


Relative  velocity,  bullet  strikes  balloon, 
429. 
man  and  carriage,  432. 
train  passes  carriage,  430. 
Resistance     and    horse-power    of  ship, 
135- 
and  speed  of  train,  70. 
and  time,  projectile  stopped  by  sand- 
bank, 490., 
overcome  by  bicyclist,  128. 
overcome  by  railroad  train,  i. 
overcome  raises  triangular  mass,  286. 
to  passage  of  warship,  136. 
Westinghouse  brake,  131. 
Rest,  body  moved  from,  166. 
body  moved  from,  by  constant  force, 
410. 
Resultant  of  forces,  at  comer  of  equilat- 
eral triangle,  292. 
square,  306. 
acting  along  sides  of  square,  231,  254. 
on  three  sides  of  rectangle,  299. 
on  two  sides  and  diagonal  of  square, 
232,  233. 
Resultant,  four  forces  at  a  point,  182. 
of  couple  and  force,  308,  311. 
three  forces  at  a  point,  181. 
three  forces  act  on  bar,  287,  289. 
two  forces  at  right  angles,  174. 
two  horizontal  components,  175. 
two  persons  lift  a  body,  177. 
Revolutions  and  horse-power,  88. 

of   fly-wheel     before    stopping,    159, 

163.  164. 
of  pendulum,  480. 
Rifle  projects  shot  horizontally,  height 

above  ground,  475. 
Right-angled   triangle  weighted  at  cor- 
ners, center  of  gravity,  339. 
Ring  held  on  hoop  by  string,  reactions, 
244. 
on  cord,  equilibrium,  243. 
River,  boat  crossing,  landing-place, 428. 
Charles,  horse-power  of,  gis. 
velocity  and  course  of  boat  crossing, 
427. 
Road,  wagon  drawn  up,  pull,  204. 

Sails,  action  of  wind  on,  239. 

effective  pressure  of  wind  on ,  240. 
Saint  Mary's  P'alls  Canal  Lock,  horse- 
power to  pump  out,  1 14. 
Sand  bank    stops    bullet,  distance    of 
penetration,  150. 
stops  projectile,  resistance  and  time, 
49-.. 
Screw,  differential,  mechanical  advan- 
tage of,  35. 
pressure  on  head,  34. 
Screw-jack,  weight  lifted  by  given  force, 

.67. 
Screw-press,  pitch  of  screw,  32. 
ratio  of  weight  to  power,  33. 
Shaft    emptied   of    water,   time   neces- 
sary, 122. 
loaded,  horse-power  to  turn,  398. 


152 


ME  CI/ A  N/CS-PR  OBLEMS. 


Shaft,  revolving,  heat  generated,   399, 
400,  401,  402. 
sunk  in  chalk,  work  done,  14. 
Shafting  rests  on  two  supports,  pull  to 

raise  one  end,  281. 
Shears   lifting  gun  at  works  of  Mary- 
land Steel  Co.,  224. 
forces  in  legs  and  stays,  225,  226. 
Shell  bursts  into  two  fragments,  original 

momentum,  261. 
Shield  pierced  by  ball,  loss  of  energy, 

147. 
Ships,  speed  of,  140. 
Shot  discharged   from  gun,  recoil,  494. 
discharged    from  gun,    force  of  pow- 
der, 492. 
work  done,  153. 
fired  from  gun,  recoil  up  incline,  491. 
penetrates  target,  distance,  148. 
projected  horizontilly,  height  of  rifle 
above  ground,  475. 
Sinker   in  running   water,  equilibrium, 

235. 
Six-masted    schooner,  horse-power    to 

unload  coal  from,  78. 
Sledge-hammer  swung  by  blacksmith's 

helper,  165. 
Slipping    position    of     ladder    against 

rough  wall  and  ground,  377. 
Slope,     wagon     rests     on,    equivalent 

forces,  202. 
Smooth    surfaces,    friction    coefficient, 

350  to  351. 
Solid  cylinder  within  hollow  one,  center 

of  gravity,  346. 
Space   passed   over  by  body  projected 
down  incline,  464. 
sliding  down  incline,  422,  423. 
moving     weights     on    threads    over 

pulley,  459- 
ice-boat,  and  velocity,  412. 
Speed  of,  belt  for  given  horse-power,  65. 
driving  pulley,  64. 

locomotive,  in    given    distance    and 
time,  69,  73. 
up  a  grade,  70. 
maximum  of  locomotive,  57. 
ships,  140. 
Sphere  between  wall  and  incline,  pres- 
sures, 213. 
held   on  incline    by  string,  point   of 
attachment  of  string,  247. 
Spoke  or  wagon  body, push  on,  relative 

effects  of,  267. 
Sportsman  shoots  running    deer,   aim, 

426. 
Square,  and  equilateral  triangle,  center 
of  gravity,  338. 
center  of  forces  acting  in,  297. 
components  of   velocity  along  diag- 
onals of,  424. 
forces   along  sides   of,  resultant,  231, 
234. 
sum  of  moments,  296 
forces   at  corners,  force   to   balance, 

2CO. 


Square,  forces  at  comers,  resultant,  306. 
forces  on  two  sides   and  diagonal,  re- 
sultant, 232,  233. 
moment  of  forces  acting  in,  297, 
Stable  equilibrium,  347. 
Steam  used  in  machine  discharging  pro- 
jectiles, 133. 
Steam  crane,  work  done  and  wasted,  77. 
Steam  engine,  efficiency  for  coal  burned, 
9'- 
horse-power  for  coal  burned,  92. 
horse-power  of,  94,  95,  0,  97. 
pressure  on  piston  guides,  194. 
pressure     lor    engine    of    given  horse' 

power,  71,  85. 
Steamsliip,  horse-power  of,  140. 

resistance  and  hoise-power,  135, 
Steamer,  apparent  and  true  velocity  of 
wind  as  seen  from,  434. 
engines  reversed,  time  to  stop,  439. 
Steelyard,     length     of     pound-graduar 

tions,  317. 
Stick  rests    on    nail   and   smooth  wall, 

equilibrium,  326. 
Stone  blasted,  vertical  height  and  hori- 
zontal distance,  469. 
Stone,  block  of,  carried  by  two  men,  283. 
dragged  on  rough  ground,  weight  of 

stone,  355. 
dropped   from  balloon,  height  of  bal- 
loon, 419,  420. 
from   moving  train,  horizontal  dis- 
tance, 468. 
into  well,  depth  of   well,  415,  417, 
418. 
force  to  pull  wheel  over,  211. 
just  slides  down  hill,  friction,  357. 
moving  with  decreasing  velocity,  dis- 
tance traveled,  4f  6. 
on  ice,  push  to  move,  351. 
projected     upward,     time     to     reach 

ground,  414. 
rising,  meets  one  falling,  initial  velo- 
city, 416. 
skimming  on   ice,  distance  traversed, 

411. 
thrown  from  tower,  time  and  distance 
reaching  ground,  467. 
Stream,  energy,  possessed  by,  12. 

horse-power  of,  60. 
Stresses  in  bridge,  254. 
frame     of    car-dumper    at    Carnegie 

Works,  222. 
King-post  truss,  255. 
roof  truss,  253. 

tackle  and  boom,  187,  188,  228. 
tackle  and  boom  hoisting  coal,  189. 
triangular  truss,  186. 
Stretch  of  string,  169. 
String  and  wall  holding  inclined  beam, 
tension,  320. 
holding  ring  on  hoop,  reactions,  244. 
holding  beam  against  wall  and  ground, 

tension  and  pressure,  319,  322. 
holding  pole  against  wall  and  ground, 
tension  and  pressure,  325. 


ALPHABETICAL    CLASSIFICA  T/OA^. 


153 


String,    horizontal,    holding    weighted 
rod,  tension  and  thrust,  184. 
inclined,  holding  hinged  rod,  tension 

and  thrust,  183,  184,  321. 
over   pulley  holds    body   on   incline, 

2^8. 
weight  on  revolving  on  table,  time  of 

one  revolution,  482. 
weight    sliding  on,  position  and  ten- 
sions, 242. 
Strings,  three  meet  at  a  point,  176. 
two,  from  point  holding  rod,  tensions, 

198. 
two,  support  weight,  tensions,  180,  igg, 

215,  216,  220,  229.  ■» 
two,  supporting  beam,  tensions,  igi. 
Strut,  inclined,  supporting  weight,  held 

by  horizontal  rope  to  wall,  184. 
Supports,  reactions  of  weighted  rod  on, 
291. 
rod  resting    on    two,   pressures    on, 
268. 
Surface  level    of  lake  lowered,  horse- 
power, 1 19. 
Suspension  foot-bridge,  forces  in  cables, 

251. 
Swinging  ball  on  end  of  string,  158. 

Table,  block  dragged  on,  reaction,  353. 
of    planing-machine,    work    done  in 

moving,  371. 
mass  dragged  along  by  falling  body, 

weight  moved  on,  coefficient  of  fric- 
tion, 352. 
.    weight  on,  just  slides,  friction,  356. 
Tackle  and  boom,  stresses  in,  187,  189, 
228. 
pull  to  raise  weight,  39,  40. 
Tacks,  three  in  isosceles  triangle,  pres- 
sures on,  197. 
Tank   pumped    out   by   ten    men,  time 
required,  12  t. 
water  level  lowered,  work  done,  25. 
Target,  distance  of  penetration  of  shot. 

Tennis-net  poles,  inclination  of,  due  to 

friction  in  ropes,  370. 
Tension  and  position  of  weight  sliding 
on  string  from  two  points,  242. 
and  pressure,  beam  held  against  wall 
and  ground  by  string,  319,  322. 
rope  holds  beam  against  wall,  201. 
and    thrust,    horizontal    string  holds 
weighted  strut,  184. 
inclined    string    holds   hinged  rod, 
183,  184. 
and  wind    pressure,  balloon    held  by 

inclined  rope,  190. 
in  bolt  pulled  up  by  spanner,  382. 
in  cord  passing  over  pulley,  cord  mov- 
ing. 460. 
in  inclined   chains  supporting   beam, 

307- 
in  picture  cord,  219. 
in  rope  around  post,  385. 


Tension,   in   rope,   engine    draws    cage 
from  coal-pit,  461. 
in  string,  conical  pendulum,  479. 
in   thread,  body   on    table   moved    by 

hanging  body,  449,  450. 
threads  over  pulley  supporting  moving 

weights,  462. 
tie-rod  of  roof  truss,  252. 
tight  and  slack  sides  of  rope  drive ,  388. 
two  strings    from   point  holding  rod, 

19S. 
two  strings  support  beam,  191. 
Tensions  in  drivmg-belt,  63. 

pulley  cord  transmitting  horse-power, 

two  strings   supporting   weight,    180, 
199,  215,  216,  220,  229. 
Thanksgiving    Fire  at  Boston,  coal  re- 
quired for  engine  at,  162. 
Thread    connecting  body  on  table  and 

hanging  body,  tension,  171. 
Thrust  in  connecting  rod,  moment  about 

crank  pin,  329. 
Tides,  horse-power  of,  132. 
Tie  and  jib  of  crane,  forces  in,  188. 
Tie-rod  of  roof  truss,  tension  in,  226. 
Time    and    place    of    meeting,   falling 
body  meets  risina;  one,  421. 
for    cannon-ball     fired    from   hill   to 

strike  sea,  472. 
for  stone  projected    upward  to  reach 

ground, 414. 
to  stop  steamer,  engines  reversed,  439. 
of    motion,    hanging    weight    moves 

weight  on  table,  451 ,  452. 
of  revolution  of  conical  pendulum,  481 . 
Tool-cutting      shaft,     horse-power    ex- 
pended on,  125. 
Tower,  stone    thrown   from,  time    and 

distance  reaching  ground,  426. 
Traces,  pull    on,  horse   draws   load  up 

inclina,  359. 
Tram-car,  average  pull  on,  43. 

kinetic  energy  of,  145. 
Trap  door  held  open  by  chain, forces  in 

chain  and  hinges,  330. 
Trapezoidal    wall,    force    to    overturn, 

.327- 
Train  ascending  grade  by  momentum, 
initial  velocity,  456. 
*  pull  on  draw-bar,  445, 
horse-power  of,  at  given  speed,  54,  56. 
force  exerted  by  engine  to  move.  444- 
moving,    brought    to     rest,    distance 

traveled,  409. 
moving     on      horizontal,     work    ex- 
pended, I. 
moving,  velocity  increasing  uniformly, 

408. 
passes  carriage,  relative  velocity,  430. 
pull  from  speed,  139. 
rate   of,    judged   by  angle   of   fall    of 

raindroDS. 431. 
retarded    by  uniform  resistance,  time 

to  reduce  speed,  437. 
resistanca  and  speed,  70. 


54 


MECHANICS-PROBLEMS. 


Train  rounding  curve,  pressure  on  rails, 
47f>,,477-        ,     ,. 
running  down  incline,  130. 
stopped  by  brakes,  coefficient  of  fric- 
tion, 438. 
to  stop  in  given  distance,  22. 
Trains   meet,    velocity    after    impact, 
497- 
moving  towards  each  other,  distance 

apart,  407. 
two  on  parallel  tracks ,.velocity,  435. 
Trench  „  horse    lifts   earth   from,   work 

done,  16. 
Triangle,  equilateral,,  center   of  forces 
at  corners,  293,  294. 
resultant  of  forces  at  comers,  292. 
forces   along  sides,  sum  of  moments 

about  base,  295. 
lifted  by  force  at  corner,  pressure  at 

base,  340. 
right,  weighted  at   corners,  center   of 

gravity,  339. 
weighted  at  vertex  rests  against  wall, 
230. 
Triangular  platform    loaded,   pressure 
on  supports,  302,  303. 
mass  raised  by  overcoming  resistance, 

.       ^^^ 

Trigonometric  functions,  page  134. 
Tripod,  compressive  forces  in  legs,  223. 
Trucks  on  inclined  tramway ,  weights  of, 

2og. 
Truss,  255. 

bridge,  stresses  in  post  of,  323. 
king-post,  stresses  in,  255. 
triangular,  stresses  in,  186. 
Turbines  at  Niagara,  actual  horse-power 
of,  105. 
Whiting  Paper  Co.,  Holyoke,  Mass., 
horse-power  of,  118. 

Underwriter  fire-pump,  work  done  by, 

109. 
Uniform  bar  weighted  one  end,  weight 
to  balance,  276. 
beam     hanging     by    one    end,    foot- 
pounds to  raise,  27. 
beam  weight  at  one  end  to  balance, 

276. 
circular  disk  punched,  center  of  gr^v- 
,    ity,  341- 
lever,  fulcrum,  279. 
rol  on  two  supports,  pressures,  332. 
Unit  values,  page  134' 
United    States    warship   Columbia,  re- 
sistance to  passage,  136. 
Unstable  equilibrium,  347. 
Useful  horse-power  of  water-wheel,  gg. 

Velocities    and    directions,   two  bodies 
meet  at  right  angles,  500. 
and   energy,  one  body    overtakes  an- 
other, 499. 
Velocity  after  impact,  freight  train  runs 
into  passenger,  497. 
after  impact,  two  balls  meet,  496. 


Velocity  alorg  diagonal  of  square,  com- 

ponents  of,  424. 
and  course  of  boat  crossing  river,  427. 
and    distance,    cars    start    from    rest 

down  incline,  458. 
and  space,  ice-boat  started  from  rest, 

412. 
angular  of  fly-wheel,  436. 
ball  projected    on   rough    horizontal 

plane, 440. 
boat  moves  away  when  man  jumps  in, 

493- 
body     moving     horizontally     against 

friction,  457. 
body  slides  down  incline,  465. 
components   of,   body   sliding    down 

incline,  425. 
decreasing,  distance  traveled  by  stone, 

406. 
height   for    bullet    to    fall    to   attain 

given,  151. 
initial,  rising  and  falling  stones  meet, 

416. 
initial,  train  ascending  grade  by  mo- 
mentum, 456. 
magnitude  and  direction  of  body  pro- 
jected upwards,  463. 
moving  body  under  constant   acceler- 
ation, 405. 
moving  trains    increasing    uniformly 

408. 
relative,  bullet  strikes  balloon,  429. 

man  and  carriage,  432. 

train  passes  carriage,  430. 
to  give  certain,  equivalent  to  raising 

through  vertical  height,  144. 
two  trains  moving  on  parallel  tracks, 

435- 
water  in  pipe  and  shut  off  suddenly, 

487. 
wind,  apparent  and  true  from  moving 

steamer,  434. 
with    which   stone  projected    upward 

strikes  ground,  414. 
Vertical   and   horizontal  components  of 

inclined  force,  178. 

Wagon  drawn  up  road,  pull,  204. 

rests  on  slope,  equivalent  forces,  202. 

weighing  one  end  of ,  at  a  time,  273. 
Wall,  beam  in  hemispherical  bowl  rests 
against,  equilibrium,  245. 

smooth,  beam  held  against  by  rope, 
201. 

triangle     weighted     at     vertex     rests 
against,  230. 
Water  pumped  from  mine,  115. 

shaft,  horse-power  required,  123. 
time  required.  122. 

raised     by    engine,     horse-power    re- 
quired, 53. 

by  windmill,  horse-power  of  mill,  48. 

from  mine,  work  done,  4,  24. 

running,  sinker  suspended  in,  equilib- 
rium, 235. 

stream  flowing,  energy  of,  12. 


ALPHABETICAL    CLASSIFICATION. 


55 


Water-fall,  cubic  feet   required   to   de-    | 
scend,  102. 
height  required,  101. 
horse-power  of,  119. 
Niagara,  horse-power  of,  105,  106. 
Water-motor  drives  dynamo,   kilowatts 
generated,  62. 
Pelton,  tested  by  friction  brake,  397. 
Water-wheel    horse-power  of,  99,   100, 

104. 
Water   flowing  in   pipe   suddenly  shut 

off,  pressure,  486. 
Wedge,  rough,  angle  of,  372. 

used  to  set  up  pump  plunger,  374. 
Weight  between  two  pulleys,  deflection 
of  rope,  41. 
falling,  works   pump,   gallons  lifted, 

26. 
falls  on  pile,  force  and  impulse,  489. 
horse-power  to  raise,  46,  47,  48. 
in  hemispherical  bowl  held  by  string 

from  rim,  forces,  248. 
lifted  by  falling  weight,  42. 
lifted  by  men,  work  done,  5. 
lifted  by  screw-jack,  167. 
lowered  by  rope  around  spar,  force  to 

control,  386. 
moved  on  table,  coefficient  of  friction, 

352- 
of  body  in  moving  balloon,  433. 
of  lever  by  balancing,  278. 
of  person  increased  and  decreased  on 

elevator,  455. 
of    stone    just     dragged     on     rough 

ground, 355. 
on  inclined  planks  held  by  string  over 

pulley,  210. 
on  string  over  pulley  holds   body   on 

rough  incline,  366. 
on  string  revolving  on  table,  revolu- 
tions per  minute,  482. 
on  table  moved  by  weight  on  string 

over  edge, 450, 452. 
pull  to  raise  with  tackle,  ^9,  40. 
raised  by  differential  pulley,  36. 

drum  and  gearing,  38. 

fixed   smooth   pulley,  force  neces- 
sary, 28. 

wheel  and  axle,  37. 
sliding    on    string  from   two   points, 

position  and  tensions,  242. 
sliding  on  thread  fixed  to  rod   free  to 

turn,  equilibrium,  348. 
supported   by  component  forces    on 

incline,  207. 
Weights  held  on  two  inclines  by  string 

over  pulley,  363. 
Well,  depth    of,  by  dropping  stone   in, 

415,  417.  418. 
earth  lifted  out  of,  work  done,  13,  15. 
water  level  to  be  lowered,  foot-pounds 

necessary,  8. 
Westinghouse  brake,  resistance  of,  131. 
Wheel     between     inclines,    couple     to 

turn, 376. 
force  to  pull  over  a  stone,  211. 


V/heel  and  axle,  mechanical  advantage, 
168. 

v/eight  raised,  37. 
Whiting  Paper  Mills,  number  of  paper 

machines  driven,  118. 
Width  of  driving-belt,  66. 
Winch  handle,  power  to  be  applied,  38. 
Wind, action  of,  on  sail,  239. 

apparent   and  true  velocity  of,  seen 
from  steamer,  434. 

apparent   direction  of,  from  moving 
train,  433. 

effective  pressure  of,  on  sails,  240. 

force  of,  against  bicycle  rider,  127. 

pressure  and  tension,  balloon  held  by 
inclined  rope,  190. 

starts  ice-boat  from  rest,  velocity  and 
space,  412. 
Windlass,  rope  around  drum  of,  384. 

worked  by  horse,  force  exerted,  31. 
Wire  and  circular  disk,  center  of  grav- 
ity, 349- 

circular,  bead  on,  172. 
Wires,  two  inclined,  supporting  weight, 

maximum  load,  218. 
Wood,  block   of,  dragged   on  table,  re- 
action, 353. 
Work  done,  b(jdy  sliding  down  incline, 
17-         .  . 

engine  raising  water  from  mine,  4,  24. 

falling  weight,  26. 

fire-engine,  161. 

horse  raises  earth  from  a  trench,  16. 
from  a  well,  13. 

horse  walking  round  a  circle,  force  ex- 
erted, 31. 

lifting  materials  for  monument,  9. 
weight  and  rope  from  depth,  45. 

lowering  water  level  in  cistern,  25. 
in  well,  8. 

men  lifting  weight,  5. 
working  at  given  rate,  6. 

moving     table     of    planing-machine, 
37  J  • 

overcoming     resistance    of    train    on 
horizontal,  i,  2. 

pulling  tram-car,  43. 

pumping  engine  raising  water,  4,  24. 

punching  hole  through  iron  plate,  3. 

raising  beam  suspended,  27. 
earth  out  of  well,  13,  15. 

raising  material  from  depth,  45. 

sinking  shaft  through  chalk,  14.  . 

stretching  elastic  string,  169. 

Underwriter  fire-pump,  109. 

weight  falling  lifts  another  weight,  42. 

winding  up  chain,  10. 

winding    up   chain   and  weight   with 
capstan,  ii. 
Wrench,  tension  in  bolt  pulled  up  by, 
382. 

Y  braces  and  posts,  stresses  in,  222. 
Wrought-iron     plate,     hole     punched, 
work  done,  3. 

shaft,    metal     removed     per     horse- 
power, 125. 


^   OF  THE 


NOV  12 1918 


DtO 


\^\^ 


FEB  23 1920 
NOV  2  192® 


"*N  201911 
JAN  a   1923 


SEP  6    1926 
CT  IS  1926 


APRl.a81932 


^Wfi 


,>^/^    ■  • 


ri 


^«» 


'^r'*^:f 


!    1  : 


iiillMiHniiiiqii 
iiiliiilliriiiiHiiiililii 


\\mm 


m 


!   mi 


I 


iill 


m 


m 
III 

iljlil  I  i  |i|<|il:l 
li'llillilli! 


iiii 


liilllj!: 


iff  n     I 

I   I  In 
!  i !  I 


i 


^'  ill 

ill 

iill    i 

iiliiiiliii  ■ 
ill 

ill 


mm 


f 


iiiiiii 


Mi 


I 


]  I 


ip      P        Pi 
'      pi         I  I 
ill  I 


lii  ir 

!"  i  Ml  ' 

i  11  n  ! 


1     ] 


